How appropriate. You fight like a cow.
On the genestealer's turn, he moves a model 6 spaces and ends up directly in front of a space marine. He did not have an addtional Action Point left to attack with.
On the space marine's turn, does he have the option to shoot at the genestealer? Or must he attack with Close Combat since that is possible? Does it specify this in the rules or FAQ?
Yes, point blank shots are fine. This game does not have any 'locked in combat' type of rules. The rules don't prohibit it. The Space Marine also could take a close assault action if he was feeling suicidal.