Question 1: You either A: have a fake half life key or B: put the key in incorrectly. If you did put it in incorrectly simply reinstall and put in the right password when it asks for it. You could also try using a CD-Key changer but i dont recommend it becuase most of them are key logged(meaning they will steal your cd-key).
Question 2: LAN means Local Area Network, if your not on a router then you dont have a Local Area Network, if your not in a network you cant play lan. If you are in a network just start a game in LAN and ask somone on your network to join you(they must have same half-life and esf version as you). They should then select LAN and your game should be visible to them if they are on the same network.