If for example my chip has been loaded with double-spent Dollars without my knowledge, would I be denied that money at the next reset or not?
Good question. They're the Mint, so I bet they'll handle that the same way they handle somebody showing up at a bank with a bunch of counterfeit $100 bills: they'll ask you where you got them, and either throw you in jail (if they don't like your story) or tell you to be more careful about who you deal with and maybe direct you to some technology to help detect counterfeits in the future. But they won't let you deposit them or trade your bad money for good.