Your slow boot could be several things, but looking at the power supply first is a good call. Get a multimeter and schematics and check all your voltage outputs, especially the 5V. It may slowly rise and that's a problem. Might need a ps rebuild or maybe just new capacitors.

Regarding the coil, if it's locked on, it's likely a transistor. You can often test a transistor to see if it is shorted on with an ohmmeter. Just compare the resistance reading on the unknown transistor to an adjacent know working one. Just be sure and do this test with the board unpowered.