I think you mean "up A tree." It means that you are cornered, and have no way out of a situation. The image is from hunting, when a dog would corner an animal up a tree. The animal could not come down, but it could not escape, either. "Up a tree" means you have no solution in sight.
It means that she is crazy- mad like a dog.
It's not an idiom, it's a joke. And it's "make like a tree and leaf" -- it's a pun.
No, it is an idiom because it doesn't seem to mean anything unless you know the meaning.
An idiom
A mistake or false
It means things are trustworthy.
say no to it
To make a mistake
About to be sold, or given up.
To keep struggling and not give up.
flatter her to get something
It means shut up!