I have played this game very little. This is 20-40 live hand.

I open utg 6 handed with 2c3h4sXX. Guy who is probably way better than me 3 bets from cutoff. Folded to me. I called but think maybe I'm supposed to 4 bet?

I draw 2 and catch 5h and a brick. Opponent also draws 2. What do I do now? I bet, he called and we both drew 1. I caught a brick. Now what?

I think I didn't even spell the game right

Definitely not a 4b OOP, flat is fine. Betting flop is okay considering the 5 is a decent blocking card, but I often check OOP to avoid having to put in extra with an unimproved three-card. If you bet you can draw one and sometimes get a fold on the turn if your opponent is still drawing two, but your 3-card is very good and you can draw two to improve your chance of making a badugi without hurting your chance for the deuce side very much, and depending on player I'd do that on the first draw some of the time.

When your opponent draws one he probably has the same kind of hand as you, a three-card badugi and another low card, and it's unlikely he'll fold the turn. UI I think the turn has to be c/c, and if you brick the river you will have to decide based on past showdowns if your three-card and whatever your low is are good for half often enough to call. Pure bluffing in these spots, say you catch 6h, is not going to earn you anything vs the kind of hand he probably has. Expect a call from a good UI 3-card with any kind of bluff catch low.

Players are very showdown-bound in Badeucey when both draw on the last round so your river bets should mostly be for value. Value can be quite wide in this precise situation since your bet didn't get raised on the flop, meaning your opponent probably doesn't have a badugi heading into the turn.

I did check call turn and caught 8d. So I think maybe river is only street I horribly butchered.

I think either spelling (baduci or badeucy) is fine. (I guess spelling it whichever way the casino where you play it spells it is the way to go).

I don't know the answers to your questions. Sorry.

Buzz

What did you do on last draw? How many did he take?

He drew one also and then I check called

Yeah you have 8 perfect both ways. Either bet or x/r. Your hand is way too strong to check call

I also think four betting pre draw is fine here and with more players calling you definitely should.

I know. I was planning to check raise and then got scared.

4 betting pre heads up OOP would be bad, calling protects your range. River is too weak as jonboy said. It's important to note you should showdown 234 100% of the time heads up

You can c/r/c river vs pat, but if both are drawing you shouldn't risk the river checking through so I'd bet/call. You are rarely getting scooped with this precise hand vs a draw but similarly rarely are scooping when raised or 3b, so b/3b or c/r/4b not earning anything and hurt extra when scooped.