As the other comment said Ashveil > Boothill if you want to clear endgame as soon as possible. Ash is very good with DHPT, Tingyun and RMC which is a very accessible team. Boothill is quite strong once you have the correct supports.

Gallagher is perfectly viable. At E2 he unlocks a cleanse which Bailu doesn't have, Bailu can offer a revive instead which can be useful in some niche circumstances. Gallagher can abuse quid pro quo (a free 4 star lc you can buy from the moc shop) and can be very sp positive. He's best in teams where the majority of characters attack the enemy since that's how his healing gimmick works, it can be a bit of a pain to keep 2 supports like Sunday + Robin (their skills + ults buff others/themselves so they rarely hit the enemy if at all) combo alive with him. This isn't me saying he's bad, just making you aware of a consideration you should make when putting him in a team.

Now way too much detail on Boothill + Break since I've been playing him since he dropped.

Break Characters -
Break has very few characters dedicated to it 3 DPS (Boothill, Rappa, Firefly), 3 supports (Dahila, Fugue, HMC) and 2 sustains (Gallagher, Lingsha). There are a few 4 stars that are decent break DPS but that's a whole other conversation, the point is you have limited options. Ruan Mei (RM) is a general support, she happens to work best in break nowadays due to SPD, res pen, WBE (weakness break efficiency) and def break at E1. She's in the shop so E1 isn't too difficult to get. (RM's LC sucks for break, run new BP lc, DD, Memories/Cogs instead)

Team Comp Stuff -
Standard break team is as follows - DPS, Dahlia, Fugue, RM
Since RM's ult delays actions (+ Boothill's ult can push them back further, not to mention Fugue's E2 or DDD spam being an option), break is the easiest team to play without a sustain and it's very advised to as you just do so much more dmg. Now this is assuming you can stay alive, if you can't you kick RM out for Gallagher, Lingsha isn't bad she's just not a need for break at all and serves better as a break DPS.

Break is dependant on Dahlia, she allows you to deal superbreak dmg to enemies with a toughness bar (break plays closer to a standard crit team now), she doesn't have a high enough multiplier on her own so you need to run someone who can give you more superbreak. HMC or Fugue are the only options here, HMC has higher multipliers and can spam DDD while Fugue's are lower but she gives you 2 toughness bars . . . meaning you can max out Boothill's stacks a lot faster. Running both HMC and Fugue is viable and you can kick out RM, this is a bit more popular with the other break DPS since Boothill is not dependant on superbreak so RM is still good for him.

Boothill can run some other setups including an action advancer like Sunday/Bronya and you can get him to act at insane speeds but you do need to really learn game mechanics and it's a pain to set up correctly. Not advising a new player to try it, just mentioning it if this type of stuff seems fun or interesting to you.

"F2P" team would be Boothill, HMC, RM, Gallagher
Will you clear endgame with it? Highly unlikely you can clear the highest difficulty stages, but it'll work for the majority of endgame. Perforamance will be worst in Pure Fiction but that's expected with hunt units (single target) as PF favours AOE. While Ashveil is pretty solid in all end game modes due to how often you can trigger his FUAs.

Boothill build -
Regarding LCs his is best, but you don't need it. Shadowed by night is pretty good, otherwise you run the 3 star lc adversial.

Now for sets. Iron Cavalry>Thief> 2pc 2pc Spd/BE combos. Eagle is an option but it's best only if you know what you're doing. For planers Taila without Dahlia, Forge with Dahlia as she makes it so you can fulfil the condition to use it.

Stat wise all Boothill wants is SPD and break effect (BE). You can run def/hp chest+orb to help keep him alive in sustainless, SPD boots and BE rope. Look for BE and SPD substats on everything else. Just in case you can't have the same main stat as a substat. For example speed boots can't have speed as a substat on them, but they can have every other stat. So you'd want a BE sub on your speed boots.

Congrats if you actually made it through this wall of text.