This starts up the sprinting and doesn't cost anything but if you attack, stop, or shield you get a penalty:

What if the player always wants to be cautious? :O Automatically jumping into sprinting mode would be a little frustrating, especially if enemies were exactly five seconds of walking away.

Shielding consumes two inches of your shield bar, regardless of what your current shield health is.

Shield bash always drains 50% of your shield health.

I guess one or two of the gunslinger sets could have increased energy recharge rates. Other classes wouldn't need it so much because their primary weapons don't depend on it.

@Waffleconecake
If they always want to be cautious then they shouldn't run a straight line, just sway a bit as it only starts after five seconds of running in a constant direction

Then I bring up the alternate: what if you wanted to cover a stretch with sprinting that would take 5 seconds to cross while walking? With your idea of automated sprinting, you wouldn't be able to, since you need to wait a full 5 seconds for sprinting to kick in.

@Masterreeve
applying the bar to guns only in a game where guns originaly had infinite ammo would make people feel like it was a gun nerf and force them to use swords, which have no penalties.

Removing the reload animation from guns only in a game where using a combo attack inflicts the player with a recovery period (where they cannot attack, shield, or even move in most cases) makes people feel like it was a gun buff and force (lol?) them to use guns, which have no recovery.

The recharge rate is fast enough that you can squeeze off a gun shot fairly frequently, but at a somewhat lower DPS, and sacrificing the ability to dash or use Sprite Abilities. This would only fully affect Pure Energy-Slingers: players who only use Pulsars, Catalyzers, Blasters and Alchemers. Take a Magnus, Autogun, Antigua (if you're a pure Gunslinger) or a Gun/Bomb and switch to it when your energy is low, instead of just mashing the attack button with the one weapon (looking at you, Alchemer-Switchers and Polaris Noobs)

Another point is that having ASI on your weapon would make the bar decrease faster without increasing the recharge rate

Ah, that is a good point. However, ASI would still give the guns a higher burst damage. (You'd chew up your bar quicker, but you'd deal higher DPS than someone without ASI for those few seconds) ASI is also safer anyway, your animations finish quicker (both regular shots and charge shots) so you can move normally shortly after firing the weapon. Also, ASI isn't balanced with family bonuses anyway. ASI works all the time while family bonuses only 16% of (for normal damage guns, 25-50% for specialized damage guns)

won't get into situations in mid battle where they cannot attack for a short duration

See my above point on Pure Energy-Slingers. The recharge rate is fast enough that a player can continue shooting constantly, albeit at a less-than-optimal rate.
The basic idea is to make a few weapons a bit more powerful than the rest, but to keep them on a leash so that players can't crutch solely on them.

Charge attacks would deplete the bar by 10% upon the release of the attack

This seems fine, that's 1 second of bar recharge, anyway. Although it might cripple a gun like the Catalyzer. If the Catalyzer were to be buffed substantially, I'd have no problem with this.

Dashing still takes 50% of the bar, but possibly require a 3 second cooldown before you are allowed to dash again.

I like the concept of a double-dash, but I don't know how this would work when combined with Sprite Ability Drain: non Energy-Slingers may have enough energy to be constantly dashing, even while using Sprite Abilities.

I.E a firebolt with 18 second cooldown takes 18%, flame barrier with 26 second cooldown takes 26%, and a firestorm with 56 second cooldown takes 56%

That's correct.