I do not have a C10 and I have not used one, however in the past I have read the documentation for the board.
Just because a DC power supply is "rated" to supply a certain number of amps doesn't mean that it will actually push that rated current through the load that is connected to it. A simplified way of looking at is that the actual amount of amps that are drawn from the supply depends on the "needs" of the circuit that it is connected to.
So you should not be concerned that your 5-volt DC power supply is rated at 5 amps out. A properly operating C10 will not draw anywhere near that amount of current for the board alone.
As far as powering a coolant or mist pump, I don't know what its voltage requirements are, but you will probably be powering the pump from a separate power supply that is switched on and off by the relay.