Regulating 12 volts down to 10 volt may leave insufficient overhead for the 10 volt regulator. Also when the solenoid activates it will pull the voltage down possibly by a volt or 2 depending on the quality of the supply.
Potentially this could cause you spurious resetting of the Arduino. The power supply should be rated at a higher current than 1 amp to minimize this possibly a 2 or 3 amps unit with an output of 13.8 to 18 volts should be considered. The solenoid voltage is probably not that critical but if your supply is higher than 13.8 use a resistor in series with it or some 3 amp diodes to reduce the voltage a little.
If you have a 12 volt 1 amp supply try it and see if you get away with it but if not follow the above.