The HP looks like a good all around PC but as a gamer it allocates two much money to the i7 computer and not enough to the graphics card - resulting in less gaming performance. Also I did not see anything about the size of the PSU - a very important consideration for upgradibility and gaming in particular with the high power requirements of better graphics cards.
I suggest you look at the Dell line of gaming systems - not necessarily because they are the best choices for the price - although both HP and Dell seem about comparable in the quality of their systems, with Dell arguably still a little ahead - but to see how they allocate resources between components. Dell has systems designed for gaming - as they note in the descriptive material - including the 625: base price $799, PSU 750w, CPU starting with Athlon X2 5600; the 630: base price $899, PSU 750w, CPU starting with E8400, and the 730x: base price $1,599, PSU 1,000w, CPU starting with i7. The video cards also get larger with the price. Now you can get a good gaming system for less than $1600 with the i7 if you want it.
One key question is do you want to pay the relatively high premium, from a gaming perspective, for the i7. It will speed up any other work you do - especially video encoding if you do it, and will best future proof you system. If you are willing to spend up to $2000 you should definitely consider it. However you can still get an exceptionally fast gaming system that will handle all the candy and be future proof - relative to the CPU - probably for years to come - maybe except for all but the competitive gamers looking for the 2% to 5% edge. So it all depends on what your price/performance preferences really are.
For a gaming system the video card is king, as you probably already know. You can build a fast system around a core 2 E8400, a better system with AMD or Intel Quad PSUs - especially if multi-tasking as you say, and and finally you have the i7 at the top end.
If you have not, I suggest you read these articles about comparative systems where they build and compare low end, best bang for the buck, and high end PC's and compare results - you can see the component lists, some discussion about why they picked those components, and see comparative gaming performance tests. It should give you a better idea of the price/performance matrix and maybe a starting point - particularly about PSU selection, graphics cards, and supporting components.
First on THG this article - and go back and read the linked articles about respective systems (note that while they are build-your-own pc's - they will help educate about the respective components when you go to purchase a complete system and how price will affect performance):
http://www.tomshardware.com/reviews/core-i7-sff,2312.html
And this trilogy from ExtremeTech:
http://www.extremetech.com/article2/0,2845,2349271,00.asp
http://www.extremetech.com/article2/0,2845,2347620,00.asp
http://www.extremetech.com/article2/0,2845,2344012,00.asp
Then when you have some idea about the range and CPU, you can hone it to more information about selecting a video card - over which you should have quite a bit of control even in a factory build but customizable system:
First the best graphics cards for the money - ignoring those costing less than $200:
http://www.tomshardware.com/reviews/best-graphics-card,2362.html
And then look at the chart for the performance of individual graphics cards (note that 30 frames per second is considered playable and anything over 60 fps you wont even notice the difference):
http://www.tomshardware.com/charts/gaming-graphics-cards-charts-2009-high-quality/benchmarks,58.html
I think your CPU choice will be between the Intel Q9650 - which will provide all the power you asked for now and for the next two years - and the i7 which is top of the line, faster (but you likely wont see much if any additional speed in gaming - but in video encoding and the like), and provides a longer term upgrade path - at a price of $200 to $400 more including the upgraded mobo it requries and depending on manufacturers price.
For the Video card - well the sky is the limit - but I think you might want to consider going with the GTX 285 (about $400) or 4870x2 (about $500) today on a SLI or Crossfire mobo and add a second one later when gaming requirements increase and the price of cards fall - or if you feel rich - and you should still have enough room if you want to spend up to the high end of your $2,000 budget - you could go with 2 now. See these two compared here (a selection form the graphics charts linked above):
http://www.tomshardware.com/charts/gaming-graphics-cards-charts-2009-high-quality/compare,1345.html?prod%5B2571%5D=on&prod%5B2551%5D=on
Note that I prefer nVidia because of the superiority of the cuda feature which allows the CPU to offload some of the processing to the game card this speeding up performance, particularly on non-gaming applications. It is new and just starting to be implemented more but I think it will spread quickly.
Note that you need to decide if you might SLI or Crossfire in the future to pick a compatible mobo now and to get the right size PSU so you don't have to replace and upgrade it later.
Also note that you need to decide now if you want to be able to overlock now or in the future - as it affects some of the same things plus you need to consider it with respect to pre-built systems as some many not allow OC and for others it voids the warranty. However if you are paying a little extra to get high end components, which I am assuming above, you will get little extra performance from overclocking - except possibly better video encoding performance from OC the CPU.
Now I have probably given you enought to consider for the moment- maybe more than you wanted. But I also suggest you look at the Cyberpowerpc.com website. They are a manufacturer that provides an intermediate option. They sell pre-built systems, but give you much more control over the individual components so you have more choice, can getter better components that are more easily upgraded than the pre-built systems, and may more easily be able to OC. You might want to look at the pre-packaged systems they provide, which are pretty decent options themselves, and play with customizing them.
When you can tell us which CPU and video cards are the best fit, we can better help you configure a system. Alternatively, you might narrow down your budget range and ask for the best gaming system for say $1200 or $1500 and if, possible year or nay on the i7 CPU (since that can have a major cost impact not directly related to gaming performance). People can suggest the best system and normally feel free to suggest savings that don't affect performance when available or options to upgrade from there to increase performance if significant. Also, personal preferences and style come much more nto play with the cases. Do you just want the best gaming case or are there some particular cases you like. A convenient place to look for good options is on the Cyberpower customization list.
Other important items to address are:
1. Normally as quoted here, systems do not include monitor, O/S, keyboard, mouse, or speakers. What is in your stated, or revised budget? Normally it is best to leave these out and select to match your own preferences - or to ask about them separately as they introduce another host of variables and the O/S and monitors are addressed in separate forums.
2. What size monitor will you use? While normally not included in the budget, it does become significant in video card selection.