I need to know what the pros and cons between each of these is so I know what to upgrade my Sealed Sword to. Thanks in advance.
Avenger vs Faust
I'd suggest Faust first, if for no other reason that it helps you farm the Royal Jelly easier, and get enough jelly tokens to get a second Sealed Sword (which you can then turn into an Avenger). Both have their uses, and are very effective vs. what they're effective against.
Also, I believe that Faust swings faster than the Avenger, and it can cause curse, which makes it probably a more useful all-around weapon. On the other hand, the Avenger pwns constructs, which is nice.
Are you the type who likes slow weapons? Because if you were formally a cutter or calibur user, you wouldn't transition well when it comes to wielding a faust/avenger. Get the Khorovod instead. It is better than the Faust and Avenger, because wielding the Faust/Avenger = less challenge in the game.
It was definitely a shift to change from Caliber to Faust. Learning the timing of how to hit and not get hit back took some time.
I'll admit I've been interested in the Khorovod, mostly because it looks so big and awesome, with an impressive charge attack. I even got the Troika/Kamarin recipes so that sometime I might make myself one (hopefully a 5* version will be introduced as well). However, I have to disagree with your assertion that using inferior equipment somehow makes the game better. If you want that kind of challenge, why not just use your proto-sword and spiral brigandine? The Khorovod is far from the worst weapon, even if you're only counting 4* weapons, so it's not really a "challenge".
Get both. First you should grab a Faust to help you farm jelly better, then you should get an avenger and upgrade it to Divine Avenger. Upgrade that Faust to a Gran Faust later if you're feeling frisky.
"It is better than the Faust and Avenger, because wielding the Faust/Avenger = less challenge in the game."
It's better because it's worse! Genius!
The avenger sucks for slashing enemies, the faust sucks for charing.
The avenger is to slow for using it as a weapon, but the 3 blades that come from the charge are pretty sweet.
The charge from the gran faust is near useless, but its faster and thus more usefull in melee combat.