Setting up a rig for better gaming

Alright, I'll try to be as clear and brief as I can. January of this year I bought a new PC. I ordered one for my mom not long after. Mine is an Ultra Small Form Factor, so expanding isn't possible (I thought I was ok with that initially, but you know how it is). The other is a normal tower. The goal I'm trying to achieve is getting a better graphics card so I can play certain 2006 and earlier games better, the most demanding of which is Oblivion.

Using Oblivion as the ceiling for gaming requirements, I'll just say that I don't need 8xaa and 16xaf and various visual enhancing mods running at over 60fps with 1080p or something. Max settings with HDR, maybe 2x aa and 2xaf, running no less than 30fps in the most demanding situations, at 1240x1024 res is what I'm looking for.

However, as hinted above, the matter is a bit more complicated than simply picking the right card (more on that later). First of all I need to know if it's even possible, without spending hordes on random things, to do this. My plan was to trade out PCs, since my Mom's has a PCIe 16x slot (I think it's 1.0, but it should in general be compatible with 2.0, yes?).

However, hers has a single core whereas mine a dual, which is better for gaming. So my plan would also involve swapping out the processors. Two problems: first, do both PCs support both processors? Second, if I swapped them, would the OSs have to be re-installed? Now at last, allow me to put up the specs of both PCs to help you all be able to answer these questions.

So, after that, my next question is related to the graphics card itself. I was looking at the ATI Radeon HD 4650. Appears to have the performance I'm looking for (from comments on google and videos on youtube), and is cheap at $50. BUT, I know little about cards, so there could be a reason I can't use it (power supply/motherboard), OR there could be a better value card. There may be a slightly more expensive option that gives significant boost, or a less costly path with little performance drop. Or perhaps I'm dead wrong and such cheap GPUs would never allow me to reach my gaming goal.

Also to consider, my current GPU bottlenecks my CPU/RAM big time, but it wouldn't do much good to get a GPU that my CPU/RAM can't keep up with either (right?).

So yea, to recap, these are the questions I have.
1. Can the processors of these two PCs be swapped, and without needing to reinstall windows (because they came with Windows, so no CD/product keys)?
2. Would I need a new power supply, and if so how large, to reach my goal?
3. What graphics card would be best for me?
4. Am I simply a poor deluded fool for thinking I could game on the cheap?

If more info on either of the PCs is needed, just let me know what (and possibly how to find that info). Thanks in advance for your time and consideration.