1) I'd say there are two different definitions of lifespan for a computer. The first, I'd call "Useful Lifespan", or how long a computer can be expected to keep up with software in general. In this category I'd say the average computer has a useful lifespan of 2 to 4 years, depending on how good it was initially. If you don't game, add maybe 2 years.
The second definition of lifespan would be in the technical sense, ie: how long the components last until physical failure. Some components (such as the power supply or hard drive) are more prone to failure than others (processor or motherboard). You can expect a power supply to last around 2-3 years, a hard drive maybe 5, and processor, memory, motherboard etc perhaps 10 years and beyond.
I'm just speaking based on my personal experience here. Your mileage may vary.
2) Uhm. A budget of $600 seems too little for a gaming rig, but I like challenges. I'll try to make you a build.
I'm assuming you already have a monitor, mouse and keyboard.
: Intel Core 2 Duo E6300 1.86Ghz
: Asus P5L-MX Intel 945G Micro ATX
: Kingston ValueRam 1GB DDR2-667
: eVGA GeForce 7600GT 256MB
: Western Digital 160GB Sata 3.0Gb/s
: LG 18X Super-Multi DVD+/-RW
: Raidmax Scorpio w/ 420W (20A on +12V) PSU
Within budget! Whew, that took a lot of tweaking!
Now, pros and cons.
1) Intel Core 2 Duo E6300 processor - an excellent CPU (I have one).
2) The 7600GT video card should let you play most games in the market right now, with mid to high settings (I have one too, lol).
3) 160GB Hard Drive and 18X DVD burner.
1) Only 1GB of RAM. You can always add a second GB later when you save some money. I do alright with 1 GB of RAM.
2) Motherboard is a bit cheap and has a 2GB max memory capacity, which may be a problem in the future.
3) The power supply included with the case is of unknown quality and isn't very powerful (everyone around here will tell you case-included power supplies suck). Personally, I survive with a PSU very much like the one included in that case.
In my opinion that's the very minimum for a gaming build. Couldn't be any cheaper. My current computer is not too different from what I propose here, and I play most games (BF2, FEAR, etc) with high quality settings and good frame rate.