Re: Amumu. Don't play him anywhere but jungle. He gets eviscerated anywhere where he has to actually see an enemy face outside ganking. The reason he sees so little competetive play these days is that counter-jungling* just ruins him. That said, at lower levels where nobody knows how/when to counter jungle, he's a great pick because his ganks can be devestating and his lategame is amazing.
Re: TF. Despite being a gambler, he actually has very little RNG, although he has one of the only 3 RNG mechanics (AFAIK) left in the game, along with Gangplank and crit. He's very fun to play, even if you're not good at him, and being good at him is ALL in his R. If you want to play him, learn to read the map. Get good at looking for opportunities in various places around the map. Also, I noted you said that you needed to unlock your screen to hit with Ashe. In general, it's accepted that playing with an unlocked screen will improve your map awareness and generally being more able to play, particularly landing long range skills when you're on purple team.
Re: Karthus. Don't play him if you can't use his Q right. His passive is generally bait to newer players, it doesn't give him any more excuse to die than anybody else. People frequently joke about how Karthus is press R to win or toggle e then die, but he's entirely in his Q. If you can't land a decent % of his Qs, you're wasting your time and everybody on your team's.
*Counter jungling: The process of going into the enemy jungle in order to harass/kill their jungler and take their camps. It's worth noting that you can leave 1 creep alive in a camp and it won't respawn, so stealing when stealing a buff you should leave one of the little lizards so the enemy has to come and clear the camp before the respawn timer starts.