I am not certain that "after all" is an idiom, but it is an adverb that, according to Websters means "in spite of considerations or expectations to the contrary".
So:
Despite the fact the bus was cheaper and left earlier, he took the train after all.
The weekend turned out to be warm and sunny after all. (Despite weather predictions to the contrary).
We must put an end to piracy, once and for all.
"Hit the books" is a sentence.
"Eat, drink, and be merry" IS a sentence.
I had to drop that project like a hot potato when I found out about all the issues with it.
He told me I had won, but I thought he was pulling my leg.
Nobody wanted to do the job, so we all past the buck. Joe passed the buck.
The more honest a person is, the more he bears the brunt. This is an idiom stating the ordinary man.
Leaders like her are a dime a dozen.
Yes: "Beauty is in the eye of the beholder."
keep your chin up!
that night i gave mama a test and it was a peice of cake
The idiom "keep an eye out for" refers to watching for something or someone. An example of a sentence using the idiom would be: Jeff should be arriving soon, so keep an eye out for him.