Those relays you hear clicking might not be relays, but the solenoid itself.
When they get old they often get crap in them that clags up and arcs can effectively weld (for want of a better word) the solenoid pin to the barrel, so that when you hit the key, the solenoid doesn't throw out. If the solenoid doesn't throw, the circuit for the motor itself doesn't close so it wont spin over.
Most likely when you pulled the motor out the solenoid was dislodged.
Try hitting the solenoid with a rubber mallet then start it. As for the lights coming on, most cars do that so you can see what globes are gone on the dash display.