3) mount it to an old motherboard tray I have laying around
This would seem your best option, then mount this to a board that you could put into the cabinet. See my reasoning below.
The obvious benefit of having it in a case is that if I need to work on the computer, it's easy to just grab the whole thing out of my cabinet and throw it on my workbench. It's also convenient in that you have places to mount everything. If you mount the motherboard to a piece of wood, then you have to figure out how to mount the hard drive, power supply, and possibly CD-ROM and floppy if you are using them.
Tear the drive rack out of an old/POS case and mount it on the wood with the MB tray. And your good to go.
CPUs in cases usually run cooler due to the power supply sucking air through the case.
This isn't true. If you put it on a board and then put 1-2 fans nearby, you'll be doing the same thing. In most setups, the PSU does minimal air movement and isn't actually enough for full exhausting. The volume of air in the cab, and any fans you have that circulate ambient/outside air into it will almost definitely keep the processor and the like cool enough.
The potential cooling problems still exist, however.
Peopl regularly just open up their cases to test how well the case fans compares to ambient air. Unless you cab is tiny...like case size tiny, your not going to have an issue. Easy way to check is toss the computer inside the case into the cab, run it with Prime95 or something running for an hour and check the temps. Then do it with the back of the cab off. If you see even a 2 degree difference I'll be shocked. (add in a 120mm fan and I bet even that wouldn't happen).
Also, remember this - if you put the case INSIDE the cab, your just blowing the air out of the case into the cab anyway...so that same heated air is getting recirculated into the case later.