"In fact, the City of London is so independent that it has its own flag, crest, police force, ceremonial armed forces, and a mayor who has a special title, the Right Honorable, the Lord Mayor of London. Oddly enough, if the monarch wants to enter the City of London, she first must ask the Lord Mayor for permission"
Are there some books that comprehensively cover several aspects around the City of London?
The City of London appeared quite often in Lee Kwan Yu's memoirs
That makes much of the sword part of the ceremony, but surely it's the "red cord raised by City police" that signifies some self-authority -- what other part of the country may purposefully and physically stop the monarch and receive no investigation or punishment?
The City of London certainly has its own coat of arms. Phrasing that as "its own crest" is a bit weird. (The crest is only part, and by no means the most important part, of a heraldic achievement. The escutcheon is the important bit.)
Arms: Argent a cross gules, in the first quarter a sword in pale point upwards of the last.
Crest: On a wreath argent and gules a dragon's sinister wing argent charged on the underside with a cross throughout gules.
Supporters: On either side a dragon argent charged on the undersides of the wings with a cross throughout gules.
Yes. And local authorities in the UK routinely have grants of arms and banners/flags of them (albeit most of them are newer and less classy than the City's).
Ditto police forces: although most smaller authorities' police have merged into larger county or larger forces, there's nothing odd in principle about a municipal police force. The City's is strange, in that it's entirely enclaved within the Met, but there are plenty of oddities out there (the Mersey Tunnels Police, say).
There are several ancient cities with 'Lord' mayors, including York.
Are there some books that comprehensively cover several aspects around the City of London?
The City of London appeared quite often in Lee Kwan Yu's memoirs