Why not try a new battery? That will probably solve your problem. Its certainly easier and less expensive than replacing the starter.

You do not need a battery mounted in the car. The 'jumper' battery will serve as the car battery. However, if the jumper cables are not heavy enough gauge wire, they will introduce resistance in the starter circuit and drop the voltage to your starter. This voltage drop will appear as a weak battery and the starter may not crank the engine fast enough to start.

If your cables are getting hot, they are definitely not heavy enough wire guage. Cables in parallel will help reduce the voltage drop. This is about the same as using heavier guage wire.

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