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I had a guy fall unconscious touching cover. Can the other player spend an action again and shoot him to finish him off to dead? If yes, does my guy get cover since hes touching it? and the enemy just gets a normal roll?

I know nothing about N3, but in the previous edition once someone becomes unconscious they fall prone at the end of the order where they went unconscious. If the person who shot them still has a line of fire (they were on a roof shooting down or something) then they could of course shoot again with another order (and the model shot would receive any cover bonus it was entitled to based on the line of fire rules). If they are on the same level and falling prone removes the line of fire then you can not shoot them again with another order because a line of fire no longer exists.

Note you can only perform one "non movement skill" (such as shooting) per order. So no "I spend an order, I shoot, I shoot again as my second action". it has to be move move, move shoot (shoot move also works), or shoot do nothing.

Check the silhouette as though it were prone.

This makes chest high walls very useful in that, if you don't get killed instantly, your unconscious trooper is mostly safe behind the wall. This allows a medic or doctor to more easily get to them.

Yep, N3 uses the silhouettes for checking cover. If you were behind cover and fell, he would need to see basically just hte base height to shoot again.

If your model was at a building corner or something, with a partial base viewing, then he could still shoot at your model but with cover modifiers.

pg185 of N3 states:

Unconscious troopers automatically fall Prone unless they are a Trooper Type unable to do so

Most models will go prone. Remotes, TAGs, and such will not.