No, they are not just sitting idle, the Fed needs that money (and it needs to be liquid money) to do it's job on a daily basis, in the same way a company needs a checking account.
Gold is certainly sitting idle.
Again, that is only because it was left over from 100 years ago and we haven't gotten around to getting rid of it yet since it's an insignificant amount of money.
What is wrong with just letting the free market decide the price of Bitcoin? Why do you need the government to compel taxes from citizens in order to prop up the price?
There's absolutely nothing wrong with letting the free market determine the price for bitcoin, but I'm thinking that a plan to secure the nation's finances with bitcoin-- similar to how other hard assets have been used throughout history-- might actually make sense.
You are asking US taxpayers to buy Bitcoin instead of paying for people's health care, retirement and the military--because you want your particular choice of investment to go up in value so you can get rich.
Again, I'm rather dumfounded that people in the Bitcoin community would want this entanglement with the US government. It goes against everything that the "Bitcoin movement" ever stood for, and replaces the idea with a government special interest giveaway to some rich investors. And there's absolutely no way that a tiny number of American get rich using US government outlays without them wanting something in return, and/or being subjected to some other backlash, e.g. forcing all Bitcoin mining to be in the US, and be with "registered" miners who are controlled by government regulators.