Will a large capacitor damage a circuit
Yes
It shouldn't. It should smooth out voltage droops.
Yes, it acts more like a constant voltage source. An infinite sized capacitor would be an ideal voltage source.
It should be mounted between positive and ground. Basically, it should look like it's a parallel mounted battery. 


You can think of a capacitor kind of like the tank on most toilets. It stores extra water for the times that you need extra water (current) to flush. The capacitor in this analogy would be like a pressurized toilet tank. With pressure being voltage and the volume of water being current. When your servo turns on, you now have more temporary volume of water at line pressure to satisfy your needs if you install a capacitor. In short, more current at the same voltage.