Can you verify that C:\Riot Games\League of Legends is indeed where your game is installed?
You can also add C:\Users\USERNAME\AppData\Roaming\LolClient\ * (no space) to the exceptions list
Make sure you restore any quarantined files and if you have avast firewall to allow all connections for everything that is LoL, likewise in windows firewall if that is active (Do not have more than one firewall/active protection client running! This includes disabling windows defender!). I had like 50 or so entries for LoL in my firewall application rules which could be causing your issues as well. There is a simple clean-up tool you can run to get rid of the crap and then allow all the correct connections to LoL. Verify your firewall is set to private mode as well. After that repair the LoL client and restart your machine. You can also uninstall pando media booster if you haven't. It doesn't seem to affect my client but it may as it is not used anymore.
If this is still happening I would suggest making sure you have the latest software version and definitions updates for avast (as well as making sure it is registered and active) and you do a boot-time scan with high sensitivity. You most likely have a virus that has embedded itself into one of your drivers that is called when the game client loads and avast is actually doing its job. Clean it up and you should be solid. You may have to re-install the affected driver.
Running an sfc /scannow in an elevated command prompt never hurts either for repairing system errors.