Your first two questions are interrelated, so I can answer them both without going against eNotes policy of answering only one question per posting.
Mr. Merryweather, the pompous bank director, is essential to the story because he can explain exactly what the burglars are after. He does not want to be at the bank on a Saturday night. He seems to suspect that this is going to be a "wild goose chase." The policeman, Mr. Jones, assures him that he can have complete confidence in Sherlock Holmes.
"Oh, if you say so, Mr. Jones, it is all right....Still, I confess that I miss my rubber. It is the first Saturday night for seven-and-twenty years that I have not had my rubber."
He is talking about a rubber of bridge at his club, of course. Later, when the party is gathered in the bank vault, Mr. Merryweather, explains precisely what is at risk in this burglary.
"We had occasion some months ago to strengthen our resources and borrowed for that purpose 30,000 napoleons from the Bank of France. It has become known that we have never had occasion to unpack the money, and that it is still lying in our cellar. The crate upon which I sit contains 2,000 napoleons packed between layers of lead foil."
Arthur Conan Doyle frequently introduced skeptical characters like Mr. Merryweather in his stories. Typically they would express their doubts about the abilities of Holmes, since he was not a Scotland Yard detective but someone they regarded as an amateur. Holmes invariably had the satisfaction of proving to them how wrong they were for mistrusting him.
Some of these skeptical characters were actually policemen themselves, and they showed the usual mistrust that professional police have for private detectives. This theme has been copied in many private-eye novels up to the present time. Holmes is respected by many of the professionals who have worked with him and have benefited from his genius and his unorthodox methods. Consequently, Holmes has policemen like Inspector Lestrade coming to him for advice and also authorizing him to act in a quasi-official capacity in cases such as murder in which private investigors would normally be strictly excluded.
It was Edgar Allan Poe who first used the idea that a private individual like his Dupin could investigate a murder mystery because he was acting under the aegis of an official in the police force. Here is a significant paragraph of dialogue in Poe's "The Murders in the Rue Morgue":
“As for these murders, let us enter into some examinations for ourselves, before we make up an opinion respecting them. An inquiry will afford us amusement,” (I thought this an odd term, so applied, but said nothing) “and, besides, Le Bon once rendered me a service for which I am not ungrateful. We will go and see the premises with our own eyes. I know G—, the Prefect of Police, and shall have no difficulty in obtaining the necessary permission.”
With these words Poe established some of the conventions of the private-eye story, which Doyle copied in his Sherlock Holmes tales. The private-eye knows a high-ranking police officer, and he also has a personal motive for wanting to help someone who is in trouble, in Poe's story a man named Le Bon.
(Incidentally, it was the unarmed Holmes himself who "seized the intruder by the collar" in climax of "The Red-Headed League.")