I think I have mis-understood the purpose of your circuit, as I thought you were using the 555 as an oscillator and doing something involving some kind of frequency with your solenoid. So you are using it to send on or off states, so the solenoid should be either on or off, and not oscillating?
In which case you definitely need those decoupling caps!
1N4007 would do, just, but you'd be better using faster diode such as 1N5819 (or one of it's higher voltage cousins) or a high speed rectifier.
Yes, base to emitter, though to be honest if the "off" condition of your receiver's output is below 1.2V you needn't worry about it since this is below the transistor's turn on voltage anyway.