This could give you an insight on what the community has:
http://store.steampowered.com/hwsurvey
Personally, I suggest:
OS: Windows 7 x64
RAM: 4-6GB (I've got 4GB and i never had big troubles before)
CPU: ~3.3GHz Quad-core
GFX: AMD HD 7850 (If you have a low budget, you could even get the 6700-6800 ones, I use a 5450 and I'm able to run most games on low-mid details)
No experience with NVidia's, but the community loves the GTX660.
VRAM: 1GB is well enough. If you really want to push, 2GB.
Screen Resolution: 16:9 no matter what. Full HD is best, I personally use a 1600x900.
HDD: 750+ GB (I'm really short of HDD space with 500GB)
If you plan on playing GOOD, buy a mic. Don't expect to win as much without communication, unless it's a FFA/Arena game.
Network should give you less than 100ping most times (Max 120).
Answering the questions one by one:
I'd rather make my own, but I own an assembled one (which I personally modified, kind of a hybrid).
It doesn't take THAT much time if you're already into it, it's like a puzzle with very few pieces.
If you plan on getting a new pc, you should also get a new case ("tower") to make sure you have room for everything.
For the cooling system, the high-end pc's heat quite a lot, so you should get a liquid cooling system, or a decent amount of fans.
I really suggest Intel CPU's, but some people say AMD CPU's + AMD GPU's work well together, can't tell.
For the motherboard, just get one with all you need (Enough RAM slots, enough space for your Gfx Card, enough USB slots...)
Most new Gfx Cards come with a HDMI output (Sometimes they ONLY come with HDMI outputs, so check it out and see if you'll need a new screen).
Make sure your power source can handle all of it (700-800W should do it)
To make it cleaner:
Most (All?) the pieces have their own connectors.
You need to make sure you have the space itself to place the pieces, as some are high/long as twice as others and may not fit in a regular PC.
As an example, a GTX660 probably wouldn't fit in my pc, because it's too long.