Usually motors will require a pinion, meaning you can either buy a new one or use the one from the existing motor, assuming that they are the same bore size.

Your battery should have at least the equivalent amp rating as the ESC.

In terms of c ratings and burst ratings...higher is always better, but not often needed. I know that with my Traxxas mini rally's they require some batteries that are around at least 30c. Batteries getting hot or even warm is a good indicator that you need more current flow through. The Lipo batteries that are in my racing vehicles (listed below) never get the slightest bit warm.

From what I've found it's pretty hard to find a high C rating battery for the mini vehicles. Most will either come in at 20c or 30c. Or you could go with max amps batteries where they list their true c rating as 100c, you just pay at least twice as much.