Sure! It's all part of analyzing a game. Act like you've never seen the game before and you're somebody brought in to annotate it.
Should you examine your oppenents mistakes as well?

In many GUI's you have the choice of analyzing white, black or both sides. In my own games, I always use both sides. If you leave the opponent's side out of the analysis, you must assume best play in order to choose best response. But your opponent did not play best moves, so your best moves are against some weak moves which may not tell you as much.
I mostly analyze my losses which is where I learn what I did wrong. I may have played a quicker, stronger game in my wins, but I don't learn much when the opponent does not play his/her best, or is a very weak (beginning) player.
and the possible better moves they could have made? when going over your own games with an engine or w/e
up to now i usually just focus on my own mistakes but im beginning to think i'll get more out of my own games by analyzing my opponents mistakes as well (the not so obvious ones)