Power Supplies, Graphics Cards, PCI-e, and TDP, oh my! Someone guide me on this...

Ok, so I know that there are the official statements from video card manufacturers, but they have to play it safe. They don't know how many devices the average person has on their PC.

Still, let's limit this to video cards that do NOT have extra power connectors, and draw their power ONLY from the PCI-e slot. As I understand it, the PCI-e spec says that a system must be able to provide 75W to the PCI-e slot. Therefore:

Question 1: Should ANY video card that does NOT have an auxiliary power connector work on any PC? (assuming not too much draw from other devices)

So, I've talked about what I was looking to upgrade in this thread. Yeah, I'm limited to a single slot, low-profile solution. It's right up against the edge of the case (of course), and so I don't have the clearance for a double-height cooling solution, or even slot-and-a-half height. The card currently there measures 5/8" height for the heatsink, and 3/4" height if you count from the bottom of the PCB to the top of the heatsink. I maybe have another 1/16" to 1/8" clearance before there will be contact with the case. Fortunately, the case has a vent right where the GPU heatsink will be.

I took a closer look at the power supply (it is not replaceable - or if it is, I haven't come across any power supply of that particular long, flat shape with a higher wattage).

It is rated at 220W. It is in a Dell PC purchased in early 2014. But on the side, it lists the following information:

• 
  * 12V - 18A
  * 5V - 15A
  * 3.3V - 10A

Maximum continuous output will not exceed 220W.

There was another one I couldn't make out. Still, by my math, that comes to well over 220W. I'm figuring the 12V rail has 216W, the 5V rail has 75W, and the 3.3V rail has 33W.

Question 2: Given that the hardware (not counting the video card) consists of:

• 
  * Pentium G3220 (65W)
  * 1 500GB hard drive (typically 25W I think?)
  * 1 stick 4GB DDR3 (4W, and I may upgrade to a 2nd stick for 8GB, so we'll say 10W for safety)
  * 2 existing fans, one in the PSU (small), one on the CPU (large and extremely quiet) - (maybe under 10W total?)

I'm coming up with a maximum of 110 watts drawn, not counting a video card. Are my assumptions here correct?

I was originally looking at getting a GeForce GT 730 GDDR5, as they're available in low-profile, single slot solutions (I wish the GTX 750 was available in that form factor)

However, given my system, and assuming I can find a single-slot, low-profile version, wouldn't I be able to safely run a Radeon R7 250 GDDR5 (NOT the 250X, maybe the 250E if it's performance is on par with the 250 or old 7750)? I've seen estimates of 60-70W TDP. Or perhaps a Radeon 7750, rated at slightly less TDP? Putting me at about 180W total draw, doesn't that leave me a reasonably comfortable reserve? Or, really, ANY video card that fits physically and does not have auxiliary power connectors?

I have a feeling I might be missing something important. Still, what are people's thoughts? Would this be safe? If not, why not? The PC will be splitting its time between YouTube videos, and some FPS 3D gaming (so far Borderlands 2 will the most graphically taxing game I'll run on it, low-med settings at 1080p)

Any guidance on theory, the power draw, the assumptions I've made, etc., are greatly appreciated - especially if it helps me understand the guts of the whole thing.

EDIT: while I'm on this, how does the 250E compare to the 250, performance-wise?