There's a line in a funny song "Santa Baby" (Eartha Kitt sang it if you want to look it up) that says "I want a ring --- and I don't mean on the telephone!"
To give you a ring means to give you a telephone call, because telephones make a ringing noise most of the time (back when they were invented, that was all they could do because they had actual bells in the telephone box).
It comes from the fact that telephones all used to ring to signal an incoming call.
To keep struggling and not give up.
It means to concede defeat and stop trying.
To give someone the slip means to escape from their attempt to capture you.
The idiom give their all means to do everything you can to accomplish a task.
They like the ring.
The meaning of the phrase to ring is to give forth a clear, resonant sound. "To ring" is considered a verb. To ring can also mean to form a circle around something.
It means to feel that there is no way to go on and you just want to give up.
It's not really an idiom. It means "what are you thinking about."
RFP is not an idiom. It's an abbreviation.
It is not an idiom, it means your nose is itching.
"Have at it" means "give it a try". Another similar colloquialism or idiom would be, "Go ahead, knock yourself out", or "Go for it".