How come Abra was able to evolve in episode 22? As far as I know, pokémon can evolve when they gain enough experience or while using a specific item.
There are also some rare cases that involve specific emotions/weather.
However, I'm sure none of these was the case.  

So in episode 22 in season 1, how was it possible for Abra to evolve within the match, when he didn't even gain any EXP from it?
In the anime, evolution is a bit different to evolution in the games. For example, Pokémon do not evolve when traded and can refuse to evolve.

If you look at the list of evolution in the anime, you'll find many Pokémon that did not evolve after a battle. It's more like they constantly gain experience and can sometime decide that they want to evolve now, mostly when it's needed. Also, the idea of "levels" aren't very much used in the anime, so I don't think it really matters. It's just a mixture of emotions and experience.