I'll try to answer all of your questions as succinctly as I can, since you're asking for a lot of information.

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How viable would they be in a Gunslinger, Bombardier, or Swordsman loadout?

Gunslinger: Very Viable. They have excellent range, put out lots of shots quickly, and don't sacrifice mobility. They aren't often the fastest way to kill things (alchemers do more damage in less time, even though a full set of 6 antigua line shots can easily do more than an alchemer can do in 2) but because of their range and speed, sometimes CAN do more damage faster than other options.
When I'm using only guns, I most often take either AP or Sent. Normally using three weapons, if they're all guns at any given time, one of them is almost ALWAYS equipped. Sentenza is also my shadow gun of choice. It far outshines the biohazard and shadow driver.
I often swap between AP and Nova Driver depending on the level.
If I'm gunning in any level and don't feel like using swords or bombs, most often I'll have Sentenza, Nova Driver, and a Blight Needle. If I'm facing undead I will probably take AP over the Nova Driver or take both.

Bombardier: Very Limited. They can give you a good way to put some damage onto things as you retreat or take out turrets from a distance, but nothing you (normally) can't do better with bombs or swords. If you already have Bomb CTR from armour, you're better off with another bomb. If you need a non-bomb backup, a sword will serve you better than a gun 90% of the time. Something less specialized would be better if you HAD to have a gun as a bomber.

Swordsman: Viable. Equipped for the right level, they're great sidearms. Being quick and long range, they're awesome for finishing off that lone gremlin that's limping off, or picking off a turret from a distance, or even just flipping squitches.

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How viable are they as sidearms or main damage dealers?

Kind of ties in to parts of the first question, so not much more to say here... Short answer: Very viable for both. When I go full gunner in FSC, I am using my AP more than anything else because of the range and mobility it allows (when I do this, I take AP, Nova Driver, and a poison needle.) In Deconstruction Zones same thing with sentenza. They're extremely good against clumped up enemies because the shots tend to hit multiple enemies if they're close together.
In candlestick keep, I use AP about equally with a sword, and it's my sidearm in graveyards, in both of these cases because of how effectively it breaks blocks and crystals quickly while being very good against undead. (MAX with deadshot set with no need for trinkets.)
As a sidearm, see above. :)

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When do they outperform any other weapons? (Please give examples of when they will)

Mainly, they're compared to Alchemers because they're the most similar in terms of what type of damage options are available.
In terms of overall performance, the elemental alchemers and AP are basically equal. They both do very similar damage and though are useful for different reasons, about equally so.
AP is better for range and mobility (depending how you use it,) and Drivers can do more damage more quickly (especially using the charge to "snipe" things.) You'll see people say on the forum that for "single target damage" the AP will outperform an alchemer, technically speaking this is true, but in practice you are less often facing a single target.
Based on the math of damage per shot times shots per reload, the antigua line beats alchemers, however it's important to remember that by the time you've fired 6 shots from an antigua, you have already reloaded an alchemer or are firing a second volley.
Even against one target, if you are prepared before you attack, the alchemer charge will do more damage VERY quickly, than AP/Sent can. This puts them about equally for that too, in practice.

Regarding Sentenza, it far outperforms other shadow guns in general, in particular for fighting Gremlins. The projectiles and firing speed make it easy to beat their dodging, unlike the shadow alchemers. It is a much better offensive option than biohazard as well if you are just concerned with ending a fight quickly.
If you are fighting slimes, the same arguments apply as the elemental discussion.

In a non-ideal situation, the AP/Sentenza won't perform as well against neutral targets, though. Alchemers, pulsars, or other guns with higher per-shot damage will generally be a more effective solution if you are doing neutral damage (for example Nova Driver is pretty good against slimes, but AP is just ok.)
It's not that it does that much less damage, but it is tedious to shoot at beasts with sentenza, or fiends with AP, most of the time (not only because they dodge) or even Sentenza for constructs.

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Where is the Charge attack more useful than spamming the normal shots?

When you have space to stand still firing without much danger of getting hit, it's almost always more useful. It's generally not worth your time against one target, unless it has a lot of health (such as a giant lichen, lumbers, or even the Royal Jelly;) against a group, it can do MASSIVE damage in a large area.
In particular in FSC, some rooms specifically allow you to take advantage of this, giving you lots of space and straight lines to send a giant hawk over a group of zombies. When zombies are focused on your teammates or frozen in ice, it is a good opportunity to fire the charge through a large group since you are relatively safe then and will get very, very many hits.
(If you want numbers, at depth 22 the normal attacks will do 114 damage times six shots and hit generally at most 12 times if you are good. The charge will do 114 damage times 5 shots, then a large, wide shot that does 234 damage and will hit even just one target normally at least 2-3 times. Against a group it will likely hit EACH target 2-3 times in a larger area than most bombs.)

It will also stagger (knock down) gremlins fairly reliably. If you have a group of gremlins, and lots of space between you and them, the charge can very often take out a few of them before they get to you, and knock the rest onto their butts. In deconstruction zones, you can often use the barriers that block movement but not projectiles to do lots of damage to a group when they can't reach you with the charge, rather than picking them off one or two at a time with the normal attack.

In some circumstances, it's quite good for turrets too. If a turret hasn't seen you, you can often unleash a charge from behind it at a good distance and kill it before it turns around all the way. The initial 5 shots will fire faster than you can normally, and the last one will deal huge damage and hit a few times.
Often, it's a good idea to fire the charge off ignoring whether the first 5 shots will even hit, just for the 6th, larger shot.

Hopefully that is of some help to you.

Here's an idea: AP or Sent Charge through a Vortex might just be really, really awesome.

THIS HAS BEEN ANOTHER BOMB-RELATED POST BY DOCTORSPACEBAR

Yes, you are correct. The tightly packed enemies guarantee very, very many hits and lots of damage.

Even solo, I have used that combination without even charging my AP/Sent. just for the (many) extra hits you get.

@Jaroche: Have you upgraded an antigua to 4* or higher? They're very limited at 3*, and nowhere near potential at 4*. If you're finding yours isn't good for killing things or the charge worthless... you are probably doing it wrong.

Charge attack is great for interrupting attacks.
I use the AP to interrupt the rocket turret in the rocket room on D25 of FSC. But I have also found uses for it when using it for fun, like against lumbers.

AP works well enough for killing, though it is true you usually have to empty a full 6 shots to get a kill. Clearing a whole level with AP only would be a bit of a pain due to its mostly single target damage.

Does the Sentenza have better range than the Blackhawk? I'm constantly getting annoyed at the Blackhawk for how short the range of the regular attack is compared to my Polaris, Biohazard, and Blight Needle. I'm also not sure if I even want to get the Sentenza for my current set. My Biohazard has a slime low UV and easily outdoes my Blackhawk against slimes plus it seems to for anything normal to shadow and could easily outdo it with gremlins if it weren't for their dodge. Currently for my gunner set I'm using a Perfect Mask of Seerus with a Shadowsun Slicker and have been trying out the Swiftstrike Buckler.

And for the sake of asking it here, I'd like to know how much of a damage bonus the Plague Needle would require to OSK a trojan.

Using my front foot as 0, the Blackhawk seems to be a tiny bit above 7 tiles which still feels short to me compared to the Biohazard and Blight Needle which go a bit over 8 along with the Polaris that goes 9 due to the explosion. Other than specifically for gremlins, it just doesn't seem like an extremely useful weapon.

I guess I mean in a full party but I would like to know other sizes as well. If it's too high for me to get which is most likely, I'll probably just use my current method of taking them down. That's not too much trouble as using a Shiver followed by a charge shot and a regular attack is enough to take down a trojan with my Blight Needle.

If I can get any confirmation on that then the Sentenza will seem a bit nicer to get.

It's been tested many times, the link Jaroche posted is still correct in it's data. They just overlooked the Blackhawk when making the antiguas, but the Sentenza has the range of all the others.

Oh thank goodness, my blackhawk has been shoved into a corner somewhere.
With regards to usage of blackhawk in various areas, I find its damage quite underwhelming and its range pitiful especially when I have to snipe menders in the centre of an arena and it just won't connect, and even when it does, it takes forever to kill them. This is especially the case for T3.
In addition, I do think that the 4* antigua's charge attacks are pretty underwhelming and almost useless since they do pitiful damage and the last big bullet is hardly worth mentioning, since its hitbox is puny.
While my blackhawk is 4*, I generally favour the biohazard over the blackhawk in RJP due to greater damage and poison and general Michael Bay-ness.
Not to mention that you're grounded in the whole period of time, leaving you susceptible to attacks.

Good to hear that the sentenza is much better.

@Noogard
After my very long and lengthy suggestion post about how the plague needle (and blight) were crap relative to the blitz, the Devs (kind of) mitigated this issue with the overall gun damage boost. The plague needle's damage on FSC trojans jumped about 50%, from barely 120 a needle to now easily 190-200 a needle, and even more, factoring poison in. My current needle, which has a low damage bonus from a trinket can 100% OSK a Trojan in a 3 man party, but not a 4 man. I predict with max (or even less) damage bonus, OSKing will not be a problem. I am not particularly sure about blight though, but Toastnaut should have numbers with his VH fiend one.

PS Plague needle now does almost double what it used to on Vana.

I'll weigh in here about the Blight since it's been brought up. :P

Yup, Blight needle even at 4* is super reliable with fiend ultra (High UV and Med trinket) for taking down Trojans now.
It actually will take them down in a three player party in one charge up to the second last level (depth 26) and ALMOST is enough at 27. I suspect once it's 5* it will be strong enough. I'll see with 4 players when I get a chance.

I can with Blight. I'm not upgrading until CE goes down a bit at the next promo perhaps. :P

As far as "effectiveness with/withoug poison" is concerned, you will never make up the difference in damage solo. Where you make up for it is the extra damage done by teammates from the poison. If you are running a 4-man shiver/blitz group, one shiver, one plague, and two blitz, you should definitely be doing more damage with that than three blitz. Keyword there is "should" though...

I doubt the plague received a stealth buff. I can get the numbers but I'll need a bit of time.

Also we should create another thread, I think OP must be pretty angry as to how far we have deviated off topic :x

I've already finished collecting new damage data for the Blight, just not against vana.
I can confirm it got the same buff that all of the other guns got. It's about 10% more damage on the normal attack and 30-50% more damage on the charge compared to the old damage it did.
If my 30-50% figure confuses anyone, I can make another thread explaining what seems to have happened, and why this IS NOT contrary to what was in the patch notes, even though it said 20%. Anyhow, I'll update when I have my Plague Needle (reads: after CE dips a bit) and confirm for that one too.

LOL Veacon is right too. Even though OP got his answer, I wouldn't blame him for being frustrated that his thread got jacked.