For q1:
The windings of the solenoid have a resistance, so this comes in series with the inductor. The capacitance is distributed across the windings. If you consider a single layer of turns, it stays easy, multiple layers of turns have more capacitors that span multiple turns. See attachment.
For q2:
The equation is given is the solenoid current in a response to what? A voltage step? An impulse? A step of 1Volt or 100Volt?
Since your current is not zero at time zero, I think the only possible input is a Dirac function. (Impulse response).
Stefaan