Continuity mistake: During the film, Daniel Craig's character rides out to the house where he last saw the woman he was in love with. During the wide shot showing him on his horse riding toward the camera, the device on his wrist is missing. It is obvious if you are paying attention, as he is using the arm which is supposed to have the device to hold the reins of the horse.
Continuity mistake: When the Preacher finds Jake in his home, he is pointing his rifle at Jake right handed, but while Jake and the Colonel are searching the alien ship Jake shot down, the Preacher is pointing his gun left handed. Every other shot, he is using his right hand.
Continuity mistake: When Jake is being hit in the face, he has blood on the right half of his face by his mouth. Quite a lot running from his nose to his chin actually. The next scene he is on the ground and no blood is by his mouth.
Continuity mistake: When Jake and Woodrow escape from the cave, tons of dust from an explosion hit Jake's left side leaving him looking like he just came out of ground zero, next scene clean as a dirty cowboy can be.
Continuity mistake: When Jake throws the dynamite into the opening on the exterior of the alien's ship, he throws two bundles but only lights one. When they show both bundles of dynamite both fuses have been lit. Jon Favreau points this out in the director's commentary.
Continuity mistake: In the scene where Daniel Craig and company are chasing after an alien and arrive at a canyon while being intercepted by some bandits, Olivia Wilde can be seen getting off her horse and in an instant, on the next shot, she's got her feet on the ground with her arms in the air. This is on the Extended Blu-Ray version of the film. (01:06:30)
Continuity mistake: Earlier in the film when they come across Hunt and the other bandits, Hunt gets hit in the face and has one of his front teeth chipped and broken. But later in the film before they attack the alien stronghold when he's talking, his teeth are perfect.
Continuity mistake: When Jake rides back to the house where his wife was abducted, he pauses and looks at the ruins. The hole in the roof which was blasted by the aliens is wide open to the sky with no ridge beam. A few minutes later, during the flash back scene, he watches the aliens pull his wife through the wide open roof hole. He then steps below the opening to gaze up and there is now a large beam across the roof opening.
Answer: Don't recall seeing Liev, as for your second question, maybe Hunt figured it was nothing Jake didn't already know, or at least conclude.