Q:

Will a large capacitor damage a circuit

I have 32 servos running from a 7.5V battery rated at 2500mAh and I'm sure that it has enough power to handle the servos, but it doesn't look like it can handle all of the servos pushing at once.
I was thinking I could add a capacitor to the circuit to solve this issue, so bearing in mind that I'm very new with electronics and have only just found out that amperes are a direct result to the requirement of the circuit rather than what the battery puts out, please could you answer my questions below?

Could I put a capacitor just before the servo controller to deal with the intermittent load requirements?
Would a large capacitor damage the circuit?
Does a capacitor work just like a battery and only output the required power rather than just discharging everything it has into the circuit?
Does a capacitor have to be mounted across the positive and ground or does the positive have to go through it?

A:

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.

A:

In this situation, you can think of a capacitor as a fast charge and discharge battery in parallel to your normal battery.  
1 - maybe (see below)
2 - A large capacitor will demand a large amount of current to charge and be capable of sourcing a large amount of current for a short time.  This could damage the circuit (i.e. battery), but it depends on the circuit.  It would most likely take much larger capacitance than you are going to use to cause issues, with one exception.  When connecting a battery to a discharged capacitor, the current drawn can be extremely high as the cap charged up.
3 - In DC realm, both a battery and a capacitor will discharge what is required to satisfy Ohms law.  This means that as your load power goes up (i.e. load resistance goes down), more current will be supplied until the voltage drops.  
4 - You would mount it in parallel with the battery. 
But before you jump to capacitors, you need to look more closely at your setup.
A battery has a certain voltage and a certain internal resistance.  Say you have a cell with 4V and 0.1 Ohm internal resistance.  If you try to get 5A out, you get 5A * 0.1 Ohm = 0.5V lost with the internal resistance of the battery.  So your load only sees 4.5V.  You also may have to factor in wiring resistance from your battery to the load as further voltage drop.  At a certain point, the voltage left at the load is insufficient.  An extreme example of this is a car starting, where the starter may only see 8V from the 12V battery.  A capacitor can help by demanding less instantaneous current from the battery, and thus keeping the bus voltage higher.
It is possible that a capacitor or capacitor bank is enough to get you through the peak demands.  This assumes that your motion is not continuous on most servos and the holding power required is low.  A much more ideal solution is to size the battery such that the discharge rate is close to what is needed.  This depends on battery chemistry and is expressed in a multiple of C (battery capacity).  For example, if your battery could discharge at 3C, then you could pull 3 x 2500 mAh from it or 7.5A.
If you use a large capacitor bank to feed peeks, unless you have enough resistance between the battery and capacitor banks, you can still damage the battery.  The battery will push as much current as possible into the capacitor bank to recharge it.  Batteries are simple things and it is the responsibility of the designer to structure the load in a way that the battery is not damaged.
If you choose to use capacitor bank, you also have to realize that there may be situations when the power available is exhausted and you are pulling more than you should from the battery.  This is going to happen VERY fast unless you have an extremely large capacitor array.

