A 2.0 card shouldn't have issues with a 1.0 slot but I would avoid 2.1 card.
That CPU is going to have problems with most current games and will limit most video cards. Something cheap like an HD4650 would be a good choice. http://www.newegg.com/Product/Product.aspx?Item=N82E168...
It's quite low power and there is a good chance it will be ok on your current PSU.
The card I linked would be a good choice for that. It can even handle current games at low resolutions with reasonable settings. The question will just be whether your CPU can keep up which will vary from game to game.
If the case isn't one of the skinny one(it should be obvious if it is) then it will fit fine. 305w is uncommon outside of a Dell. They use pretty decent PSUs btw. You could use better cards but your CPU makes anything more powerful pretty much a waste unless maybe you are going to be using a really high resolution.
Not obsolete you just need to adjust accordingly. The memory type/speed combines with the bus width to give a card a certain memory bandwidth. For example a card with DDR5 on a 128-bit bus will have equivalent memory bandwidth to a card with DDR3 on a 256-bit bus.
For the HD4650 in particular there are a few models with a 64-bit bus that should be avoided while most have a 128-bit bus. Most models come with DDR2 while a handful come with superior DDR3. The one I linked for you is both 128-bit and DDR3 so it is the best of both.
Memory bandwidth is just one factor that goes into how a card performs though. The kind of processor itself is always the most import factor, except in extreme situations.
I wouldn't particularly recommend you do so actually. Even for someone with a lot of experience with this stuff looking at raw stats of a processor can only give a ballpark estimate of actual performance and of course performance is what matters in the end. So what a normal person should be looking at is benchmarks to see how the card actually performs. Your situation is complicated in that you aren't going to find benchmarks using an old single core processor so you are kind of stuck.
It's decent but with your processor I don't know if it would make much difference over the HD4650 I linked earlier. In general for that price range this is the best card around right now; http://www.newegg.com/Product/Product.aspx?Item=N82E168...
It's a very nice deal and your PSU should handle it I think but once again with your processor it wouldn't matter much. I'd just stick with the HD4650 probably and start saving up to build a new system. You can actually build something quite capable for gaming for about $300 that would be several times better than what you have now.
The memory size doesn't matter at all. More than 512mb of memory on cards of this caliber is pretty much just a marketing gimmick. It's ram speed that you want to keep an eye on and avoid DDR2 if at all possible.
That is correct. Their memory subsystems are pretty similar apart from the bus width. Twice the bus width is important but like I said earlier it is the processor that matters more. The chip in the HD4830 has twice as many stream processors running at a slightly slower clock speed compared to the HD4650.
They are the basic units that do the computations required for a video card. Sort of like each core in a CPU but in GPUs they are much smaller and less complex but there are many more. They are optimized for specific calculations common in graphic processing.