adding a gfx card wont hurt. as long as your psu is up to it. you will need a minimum of 450watts for a 98gt.
also your intel gfx is in no way near minimum spec, yes the required specs are low and you may be able to get the game installed but the chances are if your using shared ram its causing corruption.
other than that, check the admintools/eventviewer in the control panel.
your answer is helpful,though i wil say that my pc meets the req,cause my gpu is at least 128 mb what HL2 need(i am talking about original hl2,not any episode like hl2 ep 1,hl2 ep 2 etc)