RE: starter relay?
It could be.
The starter relay is located exactly where you think it is. There should be about 2, 3 or 4 relays in a row (depending on engine/options). The starter relay should be the one closest to the firewall.
If you remove the relay, there should be numbers (30, 85, 86, 87, 87a) on the bottom of the relay near each blade. If you connect a jumper from terminal 30 to terminal 87, the starter should engage (only jump for a short time, just long enough to ensure starter will turn over or not)
If starter fully engages, either the relay is bad, or there is a fault in the wiring somewhere. If you have a multimeter and another body, this will be simple. With the ignition in the START position, test for voltage across terminals 30 and 86 (PCM provides a ground via terminal 30, and the ignition switch provides positive via terminal 86).