I've tried punching people in the inn/taverns and they start lashing out like a wounded animal with weapons and spells. How do these barfights being and is there a like, drinking/eating animation?

I've only gotten the option once and it came through dialogue

You'd have to select a (Brawl) option in dialogue. Your weapons are automatically unequipped, and the opponent is set as essential for the duration of the brawl.

I've been in about 4 or 5 brawls. Just talk to some people, and they'll challenge you for money. Usually for 200 gold or so. The others I had to do during some quests.

It is in the dialogue. I was able to fight a guy who did not like dark elves and kept saying mean things about them. I was able to say I don't like your attitude, then he ask if I would like to fight about it. I kicked his tail, got 200 gold and he still talks bad about dark elves.

Beast, thanks forum.
I guess thats a no on consumtion animation?

So it seems, sadly. I sat down at the bar at the inn in Ivarstead and bought a chicken briast and an ale, then went into 3rd person and consumed them, hoping to see myself do something. But I just sat there.

In general it seems the PC's third-person interactions with the world and with other NPCs is actually reduced from that in Oblivion. Your character doesn't look at NPCs, doesn't change facial expressions, etc. I was really hoping for a bit more on that front.