Suspension should be on normal pay, unless there is an explicit clause in a contract document allowing suspension with pay.
Behaviour at a work social event can be relevant for discipline, so it might be reasonable to dismiss for fighting if managers have a genuine belief, reasonably held after proper investigation. Your argument might be that there was an incident but it was not at work, was influenced by alcohol and was out of character. You might also argue unreasonable behaviour by the other person so that you were acting in self-defence.
If one of you is dismissed and not the other, there will be considerable pressure on management to show the dismissal was not unfair.
A key issue in securing any remedy is whether or not you have more than one year's continuous service.
At the moment, there is no fee to make a claim that you have been unfairly dismissed.
Good luck.