wow.. We get a lot of garbage. Now THIS is a good question. Best solution we have is that since the Doctor is IN the machine. He can move around. She looked into his eyes and they spoke some words then they kissed. During that time between the whisper and the kiss perhaps the Doctor was able to run down from the viewing port (the eyes) down to where the machine has the lips. And reached out with his tiny hand to touch River's lips.
My understanding is that as River didn't shoot the Teselecta, River and the Teselecta are the opposite poles of the weird messed up time line. River didn't need to touch the Doctor she just needed to touch the Teselecta, and the kiss was with the Teselecta so time carried on.
Yes. What counted was contact between the two entities involved in the "fixed point" that River tried to rewrite. The fact that one of those entities wasn't who we (and River) thought it was is beside the point. As it turned out, the "fixed point" wasn't the Doctor being killed, it was the Teselecta, in the form of the Doctor, appearing to be killed. The two entities that needed to touch were, as the second answer says, River and the Teselecta.