geforce 2 is a very old card it proably run hl2 but so crappy it would be better off getting one that surpports atleast directx 8.0 (there are some for $25-$50 range) or 8.1($50-$100) and if you have the means, a directx9 card (pricy cards $100-$500)
8.0 is really no diffrent from 8.1 other than speed quality about the same
your card proably only surpports 6.0 questably 7.0 sad..
i have that exact same card and finished hl2 on it, if you put visual options on medium/low you can get about 50fps out of it, also remember to maximize the draw distance through the console, cause its annoying not being able to see barrels and stuff thats right in front of you, cheers.
It will work.
It wil Run reasonably well with everything low and low resolution, of course it depends on what cpu you have. If you have a crappy cpu then you may stuggle. Source is quite a good engine for lower end systems, but it can only go so low.
Upgrade. The only questions are:
1) How fast is your cpu. Good g-card and a crappy cpu still = bad performance. It would make it easier to help if you posted your entire system's specs.
2) How much are you willing to spend.