Delete this folder: RADS/projects/lol_launcher
Then re-open. Send me new logs when you do this if you would. please
Looks like we can call you solved. Keep me updated It'd be nice to know if you were able to successfully get in a game(try a custom game by yourself. you dont get leaves for these).
Awww you made me blush. However, I can't take all the credit we have lots of smart and talented people here at Riot.
You might take Imperial's Advice here and just wait for a few minutes after you hit play. The program may open right up. You might want to reboot your computer to refresh the whole process.
A few notes:
I am trying to assist several people in a single thread(not the most ideal environment). If somehow I skipped over your post feel free to respond again with more details and what you have tried.