I know that all batteries have a set voltage. Do batteries than have a set amount of current would flow through them if connected to a circuit?

It depends on the battery, the internal resistance and the external (load) resistance. Some batteries like NiMH, NiCd, LiPO etc. are capable of very large currents for a short period of time if you short them for example (which can result in a fire). The current in a circuit is a function of the voltage and the total resistance of the circuit (which includes the internal resistance of the battery).

No - they have rating for how much current they can supply. Some battery types will have internal resistance that limits currents (and voltage sags as a result) when connected to a circuit with little resistance. Some batteries have little resistance and can supply enough current to keep voltage up, but that can mean more quickly damaging themselves.

(remember - less resistance in a circuit makes more current flow. Think of resistance like a valve in a garden hose - the pressure in the line is like the voltage, and the current is how much water actually flows per minute. Opening the valve more decreases resistance to flow so more water flows. Same with electrons)