Zero experience here. Would it be possible to connect a simple water pump that requires 3V to the 3V3 pin of the RPI?
And in case the answer is no: Why not? What other power source would then be the easiest/cheapest to use?
Pin 1 of the header (on all models I believe, but do check yours) provides 3.3V. It doesn't give much current (it's rated to 50mA but more may be available in practice, according to people who have looked at the components.
I have successfully run two different micro water pumps (RS M100, RS M400 off this line (via a gertboard) for several days on end (while using HDMI and USB keyboard/mouse). I'm probably pushing it too hard but I haven't noticed the regulator getting warm or any misbehaviour. NB: I'm using a Pi 2, which has a switching regulator circuit; some earlier models used a linear regulator chip which would get hot under load. The limit for the 3V3 supply is no longer as clear as it was, but is likely to be higher. (official forums)
As regards back EMF -- if you have another pump in the circuit it's probably a bad idea with the M100 pump I've linked. The M400 is brushless with its own control electronics - so won't act as a dynamo in the same way. I would assume that when we prime the pump by pulling water through, we spin the motor; this has't caused any problem with either pump.
For longer term use I intend to put a TO220 (through-hole) LD1117 voltage regulator on a piece of stripboard, run off the Pi's 5V supply (2A in my case) and run the pump off that.
If you want to switch it, use something designed for the job such as the open collector relay drivers used on the Gertboard (ULN2803A).
The pump will need considerably more than the few mA the GPIO can supply. You also run the risk of destroying the GPIO/Pi if you spin the motor and generate back EMF.
Use the GPIO to switch a relay on a relay module designed to be used from 3V3 GPIO.