The Lamp of Alhazred (weapon) gives me +5 to combat checks. Does this last the entire game?
It doesn't last the entire game, but you can use the lamp for each combat check during the entire game.
You have two hand you can use to hold weapons or spells, and for each combat check you can use other items. The Lamp of Alhazred doesn't exhaust and is not discarded when used, so you may use it for each and every combat check if you like. When you cast Shrivelling you get the bonus for each combat check during the current combat, as long as you devote the appropriate number of hands (one). If you ever let it drop (say to use a Shotgun) you lose the bonus, and because it is exhausted when cast you may not re-cast it before your next upkeep.
So you have to look if the bonus applies once per use or for the entire combat; and if you may use the item multiple times per turn or not.