Add CS: Condition Zero to the above list. Some of the older CoD games like Modern Warfare should work fine and are a blast to play multiplayer. For RTS, I would suggest some of the excellent older C&C titles like Red Alert 2 or Generals; you may even be able to play C&C 3 at lower settings. StarCraft and Age of Empires 2 are also serious fun for LAN parties.
Does anyone has some suggestions for some good strategy or first person shooter games for 4-6 people; these games also have to work on a Pentium 4 3GHz with a 7600GT graphics card as the minimum.
Try Borderlands. It is a shooter/rpg with great loot. It might run on your system. The CPU meets the requirements, but the graphics card is slightly below minimum, so it may or may not work. If it will run though, it is a lot of fun for co-op.
Sins of A Solar Empire, Hearts of Iron 3 and Victoria 2.
3 People playing HOI3 in a LAN setting is quite epic.
SOASE is a killer LAN aswell, not to mention it scales to low end systems flawlessly.
Im not much for action shooters myself so I may not be much help when it comes to FPS games.