Re: Need some advice...
Well, to your questions:
1. Yes and no. In honesty, unless you already have an estimate for the laptop's repair, I don't see it being easy or cheap. Dell could/would do it, but I'm presuming at this point you're out of whatever warranty you had with them. I had an E1705 with a 7900 GT Go, and the graphics card died twice (keep that in mind, it may not be worth replacing. Graphics cards have failed a lot with these laptops). They replaced it under my extended warranty. The problem is, the replacement part is going to be very hard to find outside of Dell, and anyone who can find it will probably charge you an arm and a leg for both it and the work to replace it. It's possible but not easy at all for you to do it yourself, and I'd recommend against it. This is the yes part: see what kind of money you can get by selling it as-is. For the no part, see 3.
2. No. I'm not certain if laptop processors could be used in desktops, but in this case definitely not. Intel CPUs are never interchangeable with AMD CPUs.
3. Your big problem here is that your desktop is an older, microtower, Compaq. I'm not certain on the specs, but it's a pretty safe bet that upgrading it isn't much of an option. Graphics cards will be tough to find to fit into the proper slotting (probably proprietary) and even if that's not the problem, the case dimensions could be, the processor is an older socket type (not easily upgradeable), and the RAM is probably already maxed at 2GB.
My personal opinion is that your best option is to scrounge up as much as you can, sell what you could from the stuff you have (both laptop and desktop), and buy or build a new computer. Desktops are cheaper than laptops for the same performance, and if you're not looking for anything crazy you could squeak by with anywhere from $350-$700, depending on what games you want to run well and what kind of sales you find. I'm a huge proponent of building in this scenario, as it leaves room to buy piecewise (have a computer that runs, now, for the low end of that spectrum, and buy the additional components to get the desired performance as money comes in).
The easiest ways to save money in a case like this are to see if you can't reuse parts from the desktop. Some of the more easily salvaged pieces:
Monitor, keyboard, mouse, speakers (all of those could be obvious), as well as hard drives if they've got enough capacity, or at least enough to serve well for now, possibly (ymmv) power supplies, and the RAM you said was already an upgrade.
I could give you some more specific ideas if I knew what kind of games you want to run.