Question: Surely the Federal government would realise Goldfinger's involvement once he tried selling them his gold? That and what does China promise him for creating economic chaos in the West?
Answer: Yes, the U.S. government would realise that Goldfinger was involved. The movie's plot is deliberately unbelievable and merely intended for entertainment, not reality. Audiences are expected to employ a "suspension of disbelief." China promised Goldfinger the nuclear bomb so that he could carry out his plan. They wanted to create economic chaos in the US while Goldfinger's own gold would become far more valuable.
Question: Why did Goldfinger go through the hassle of telling the other gangsters his plan, only to kill them right after?
Answer: Goldfinger wants to keep the charade going up until the end so the others suspect nothing unusual. It is also a means of exposition to explain the plot to the audience.
Answer: I believe that Goldfinger is a showman / show off and wanted to boast for the pure hell of it. Also I think that he had to string them along so they wouldn't suspect he was about to do what he ultimately did to them.
Answer: Like all Bond villains, he wants someone to appreciate his genius, even if he plans to kill that person immediately after. How many times have villains told Bond their plans, then stuck him in a deathtrap that he manages to escape? These guys got the same treatment except for that part at the end.
Question: How did Bond know which room Goldfinger had at the hotel?
Answer: He deduced that Goldfinger was using a partner to spy on his opponent's hand, and to check his theory he went to the room with the best line of sight. Alternatively, he went (off-screen) to the desk and used his charm, which was utterly irresistible in the Bond films of the '60s, to find out where Goldfinger was staying.
Answer: He wouldn't have to sell the gold. He would keep it and become the new Fort Knox. He'd become the richest man in the world, worth billions, back in the 1960's, it was unheard of.