I've been playing Kat recently, and I've noticed that certain mids that usually drive me nuts on other champs, such as Urgot and Vlad, seem to be fairly easy to shut down. I haven't ran into any problem champs mid with her yet, but for future sake, what champs should I be wary of as Kat?

Most champs with a hard stun will give you trouble, because you have to be very careful when timing death lotus. Fortunately, most of those champs are also just as susceptable to Kat. Get practice in with Annie, Karthus, Veiger, Heimer, and Ezreal to learn how to play against them as Kat.

Ezreal is quite capable of forcing Katarina into situations that are not favorable to her and thus forcing Kat to play defensively or possibly die. You have to save your shift after level 6 for when she shunpo-ults, but on the other hand a good burster like Annie would shut down Ez mid, and probably Katarina too (I've never played AS Annie).

I play both Kat and Annie, they're definately on equal grounds due to their bursting power. If Annie misses the stun Kat kills her. If Kat tries to get cocky, Annie will kill her.

When I play against Kat as Annie, one of the worst things to run into is a fort pot + 5 heal pot Kat. The only thing to do is farm hard and let the fort pot go to waste.

kat doesn't have a direct counter mid from the standard mids
but morde and vlad are a pretty even match for her

I'd like to say that if it's AD Kat, Annie will not do well, bouncing blade hurts, a lot. I've had success with Sivir and Kayle vs Kat in mid as well as Vladimir as already mentioned.

Mordekaiser, a good Morde > Anything Mid

Never let them approach you from the side, outside their mob of creeps. If you're continually stuck making the choice between farming or harassing, you could fall into a defensive position. This is an extremely bad thing for Kat.