Is it safe to use a 12VDC Cig lighter adapter to power the PSU? The PSU takes 12V @ ~5amps. So if I get a 12V cig lighter adapter rated for 10amps and change the fuse to 5amps, would it be safe?
As long as it is regulated, yes. Sounds like you have one of those power supplies which passes the 12v straight through. An unregulated (straight through) cig lighter cable will give about 14v when the engine is running.
Alternators HAVE to supply a higher voltage to charge the battery...so your battery will give about 12.5 but the alternator has to give 13.5 to 14ish...otherwise the battery will constantly be draining.
(whatever supplies the higher voltage does the work)
The important question is do you trust the item you have to work directly off the 12-14vdc in your car.
The safe bet is to get a 12v regulator (but for 2 amps or more they get expensive).
However many people on this board have run items directly off the car power.