I can't really account for Karthus since I don't think I've ever seen one I'd deem as "threatening". However Corki is a fantastic carry, and well Anivia is my favorite hero so I'm a bit biased. For Anivia I can give you the following info:
Incredible learning curve. It requires a bit of multi-tasking and coordination. She has a high-payoff for skill however, and also for disaster. This comes in the form of a missed wall or flash frost, which leaves you quite open to dying or even worse, losing teammates because you walled them in. However, on the flipside, you become the biggest threat in the jungle when your skills improve. If you like the idea of a caster carry, play some practice games and first start by flash frosting directly into minion waves. Work on getting double hits on the ranged creeps. Then, try doing it when you are moving. Next, work on using walls. Learn how it works, how it interacts with the terrain etc. Finally, work on your combos. The first is glacial --> frost bite. You want to be able to get this combo off to a point where its basically one attack. Follow this training by doing it with a flash frost that is far away from you.
After practice games, move to real games and then try and nail the combo against human opponents. Timing is everything with her and simple practice will help you immensely. Also, if you lag, i don't recommend her. Oh, and i forgot another practice for Fog of War shots is trying killing 3-4 Ghost camps when you are in your own lane. This can only be done with a double hit flash frost, so this will help you build timing down based upon its travel speed, and not just visually.