to leave open: to delay making a decision on (S)
to lose one's cool: to get excited, angry, or flustered
to cop out: to avoid one's responsibility, to quit
This idiom is an informal version of the second definition to back out .
The noun form copout means an excuse for avoiding responsibility.
to take one at one's word: to accept what one says as true, to believe (also: to take one's word for it)
to check out: to pay the bill at a hotel or motel and then leave; to investigate, to examine (S)
to have to do with: to have some connection with or relationship to
This idiom is often used in the negative, as in the first example.