Anyone with some incite on the computer end?
Your question on CPU/GPU requirements is a bit open-ended. For example, what MAME roms are you looking to run?
An entry-level dual-core CPU, such as the Celeron G530, or an Athlon II x2, can run PSX and most MAME roms no sweat. Also, a lot of folks on this forum (me included) have purchased 3-4 yr old (and sometimes older) PC's that can easily accomplish this.
If you want to run Dolphin, you would want to upgrade to something like a Pentium G860 or an unlocked Phenom II.
Dolphin GPU requirements are a little more tricky -- it depends on more of the specifics of Dolphin that you are looking to run. The Dolphin forum has some good posts on GPU requirements that can get you started.