Your transistors appear to be wired incorrectly. The Arduino outputs should connect to the resistor (to limit current output from the Arduino) and then to the base. The emitter should be connected to ground. Your transistor symbol also shows a darlington pair which isn't at all necessary here.
See the "Using a transistor as a switch" section at http://www.kpsec.freeuk.com/trancirc.htm
for a refresher.
With respect to the compressor/tank you should treat the two as a standalone system. Normally this would be controlled by a pressure switch (example
) to automatically turn the compressor on and off and keep the tank pressure within a set range (that pressure is adjustable within the switch). The above switch also includes an "unloader" valve to release pressure on the compressor so that the next time it kicks on it gets a chance to get a sort of running start before it goes under load. That's the thing that makes a "PSSSssssh!" noise every time the compressor shuts off
To answer your question, if you keep the compressor running with the valves closed it will continue to try to build pressure until a safety valve pops or the compressor overheats/explodes/etc. If I were you I'd suggest not worrying about procuring a compressor and just make a small tank out of PVC pipe (pressurized to no more than ~20 PSI) and pressurize it with a bicycle tire pump or such. You can buy a valve stem for a car tire to use to fill the tank (just drill a hole in the side and stick it in), and for an outlet you can tap and drill one of the end caps of the tank and thread in a 1/8" or 1/4" NPT fitting which you'll later connect to your solenoids.