Dell USED to be ok, but their quality control went straight down the toilet 7-8 years ago. I did work for one guy who pushed Dell. A bunch of them all took a header one after the other. And these weren't in hostile environments, but in a high end clothing store ($200 belts, etc.). I replaced motherboards, power supplies, etc., over and over. Another customer of his was a filter factory. They bought a machine through UPS to be a shipping and receiving machine with barcode scanner, scale, etc. It came in on a Wednesday or Thursday and didn't last through Monday. It didn't run long enough for me to install the software on it. If you do buy a Dell, get the gold support. There's a good chance you're going to need it.
Cleveland Clinic leases HPs almost exclusively, and they seem to hold up very well. One of my previous employers also bought almost all HP, including network servers and also had good luck.
I build all of my PCs. It's like building an AR15, only with less physical effort. Nowadays, they're both just commodities which you can completely customize to your taste. One problem with name brand PCs is that if something proprietary breaks at a critical time, after hours, you're SOL. EVERY PC I own (other than a laptop) can (other than the motherboard) be made operational again at any time of day or night with a trip to a Super Walmart. That's usually not the case with proprietary machines. I don't know about now, but HP machines were wildly proprietary, even down to the floppy cable. The power supply was like nobody else's in the industry. Apart from the motherboard, just about anything in either of my working desktops can be replaced at 2:00am on a Sunday morning.
Regarding the hard drive from the dead machine, it's not really necessary to copy everything off of it to some other kind of media unless you just want to. As long as it's working, you can buy yourself a EIDE or SATA to USB hard drive adapter. When you get your new system, just put the drive in the adapter, connect it to the computer and you have access to all of your old data. I've saved a lot of people a lot of trouble doing that and you don't even have to open up the new computer or mess with 5-10 DVDs (if not more).