The idiom "right off the bat" means immediately or without delay, right from the start or the beginning of something. Just like a baseball is hit off the bat right away, this phrase suggests taking action or reacting promptly.
Do what ever you want to do.
Never heard of that one, but we say "take forty winks" to mean have a little nap.
it means go to sleep
It means to go faster
it means to take a chance or risk
This is not an idiom. An idiom is an expression whose meaning cannot be deduced from its elements. To go through fire and water for someone is easily understood to mean to make a special effort and to undergo difficulties for that person's sake.
A person needs to lose weight.
It means that there is not enough for everyone. It's insufficient.
It means that someone is totally surprised. They have no idea what to do or where to go, the are standing there.
"Have at it" means "give it a try". Another similar colloquialism or idiom would be, "Go ahead, knock yourself out", or "Go for it".
RFP is not an idiom. It's an abbreviation.