Call of Duty: Ghosts, stuck in "No Man's Land"

Hesh or Logan always kill the sniper for me. I have completed the mission on 3 different platforms, and it always happens this way.

Without seeing what you do, I can only guess that you get spotted or start barking before the sniper is killed. Try waiting in the house without being spotted, look at the sniper and then wait 10 or so seconds. Hesh/Logan should say something like "I got him". After, they take the shot. I am pretty sure the sniper's outline turns orange when the others decide they are going to take care of him for you.

In reference to the 2 enemies previous, I always kill the first guy and then bark to lure the other guy out. I am never fast enough to kill this second guy (Hesh/Logan always kill him for me).

General tips for not getting spotted is to stay in the grass, and attack from behind. Using the bark draws enemies to you, so maybe move around to the side/back of an enemies path after using it.