Whats up with online gaming? It's only the cat's meow!
Yeah, a comp with those specs can handle it. DSL may lag a little bit, but should suffice.
As for the card, I'd strongly suggest something with a bit more Oomph!, especially with those games. Try looking for a 9800 Pro or XT, they are discontinued and somewhat cheap, $120-140 range. A bit cheaper option is the 9600XT, starts at about $90. The prices were checked at
http://www.newegg.com, where I buy most all of my computer parts. Those cards are what HL2 was designed for.
Some more good FPS games that I like: Day of Defeat:Source (WWII FPS built on HL2, friggin awesome graphics and lighting), Call of Duty, Battlefield 2, Battlefield Vietnam, The Specialist (Matrix-style mod for Half-Life).
EDIT: This is assuming you meant Half-life 2 and CounterStrike:Source. If you are talking about the original Half-Life and CS, your onboard will probably handle it fine.