“Barking up the wrong tree”
Means: to be wrong about the reason or way to achieve something
E.g: He thinks it’ll solve the issue, however, I believe that he’s barking up the wrong tree.
“You shouldn’t look a gift-horse in the mouth”
Means: the person shouldn’t scrutinise a gift because is a gift. It came from the fact of you check if a horse is old or not when you look its teeth.
E.g: He won that gift from his friend and he is unhappy. I spoke to him: “you shouldn’t look a gift-horse in the mouth.”