Uninstall Adobe Air and then use the launcher again, it should reinstall it automatically. Try a custom game and see if it fixes your errors. If not, do the following:

Uninstall League of Legends
Uninstall Adobe Air (Again)
Reinstall League of Legends

Adobe Air will again, automatically install itself but this should also clear up whatever problems it might have interacting with the League client.

Have you tried updating Adobe Air?
http://get.adobe.com/air/

Apparently, it's a problem with LoL Replay. I started to have those crashes when I installed the new version of LoL Replay and the crashes stopped when I launched the game with the program closed.

it happesn right after i log in, so i have no idea what causes it
i try the solution of use an old adobe air and it did work, it starts happrning with the season 3 update
but i no longer have the older versions so i cannot fix it

Try uninstalling LoLReplay.

bump

same here, happens when i open summoner profile.

This started happening to me today, Also happens whenever I try to view the stats/results of a match from my ranked team's history.

Try updating to adobe air 3.6, here...

http://get.adobe.com/air/