Laxus... well first of all he isn't a huge fan of alcohol drinking but it might slip him if he gets enough of a reason why to do so.
A celebration party of the guild? Yes.
Your birthday? No.
He definitely doesn't want to skip your birthday by being drunk and useless, he wants to spend it to the fullest with you and make sure you enjoy every second of it. If you get drunk then he will make sure you're safe and spend the night with you and hold your hair if you need.

So about parties...

He will likely be like a teen Grey and Natsu are, loud and everywhere. Any drinking game he will try and likely win to impress you, if he doesn't win then he will try as many times until he actually wins and then boasts around saying he is the best or strongest.

Aftermath? As much as he is drunk and wasted he still has a bean left in his brain and won't be messy to deal with and he will try to be independent so he doesn't gross you out or leaves you alone in his mess that you have to care about him. He might need a shoulder to lean on since he is a big guy but he will be grateful for your help and next morning he will give you plenty of kisses (ntg over the top) but just enough you get the message.