Hi, really not sure about the PCIe 4x limit...
I believe if the back is cut out of the slot you can fit a x16 card in a x4 slot and it will work fine, at x4 obviously but that shouldn't be a big deal for the low ish cards you're looking at.
The professional cards, the Quadros and FireGLs won't play games terribly well due to the drivers they use. Not sure if you can "force" normal gaming tuned drivers on a pro card... Also may need to spend too much money to get a decent spec one.
If can swap the motherboard for one with PCIe x16 slot and not lose much once sold/returned existing one then that will help.
In terms of cards to get a significant leg up from the integrated graphics I would look to Nvidia GT430 or higher or AMD HD6570 or higher. If you estimate a generation = 1 series shift that's not too far away. eg GT440 = GTS250 (Nvidia 3xx series didn't really exist) or HD6450 = HD5550.
Passive HD6570 is £50~60 and likely a good bet.
Passive GT430s exist easy enough, as does Asus GT440, that's just higher clocked I think.
Also found Asus GTS450 passive for not much more than the GT440. Specs sugest 2/3 clock speed with twice the shader cores, ie 33% more power but slower, 1600 vs 1800, VRAM speed.
Lamdatek GT440 was £73 and GTS450 £87.
Asus passive GTS450 runs ~600/1200 core clocks where GTS450 more typically 800/1600 range
Sparkle have a passive GTS450
that runs full clock speed but not sure where to buy one.
I recently bought Evga GT430 for £33, noisy little brute as stock as fan runs 65% minimum once drivers kick in, no idea why. Quick BIOS mod later and fan set to 35% min, which is off when idle. Under load starts up and under the quick testing I did never got back up to noisy 65%, even under Furmark!
The HD6570 is faster than the GT430 by quite a bit and the GTS450 (full clock spoeed one anyway) is significantly faster again.
There are other bigger/faster passive cards on the AMD side if start looking, something like a HD6750 should be great for old games, if not way over-kill!