have you made sure that the energy saver pref pane isn't set up to restart the mac after a power failure, and to automatically start the computer at certain times a day? that would be my first guess.
as far as the pinwheels, did you do a fresh install of the os and apps when you got the mac? alot of time the slowdowns and crashes on used macs are from poor "house keeping" of the prior owner, and there may be so much incompatible stuff installed.
Digital Audio G4/1.467ghz, 1.5gig ram, 16x Superdrive, 256mb DDR3 AGP 6800GS, zip, 2x500gig raid0 for 1tb on sonnet tempo trio, 10.5.4