If you don't mind the paraphrase, there's a good entry in The International Encyclopedia of Secret Societies & Fraternal Orders by Alan Axelrod about Gentlemen's Clubs.

I'll keep it to your questions:

-The original gentleman's clubs were founded in the 17th c. or so- rising out of taverns and coffee houses. By the Regency era, they became, in essence, a home away from home, 'catering to what might broadly be called the Establishment'. They declined between the World Wars, and many of them lost their premises during the Blitz. While some still exist today, they pretty much faded after WW2, though reasons why vary (men increasingly socializing with their families (to an extent once considered unusual), decreasing gap between rich & poor, the advent of the weekend over the 'social season', etc).

-Membership was exclusive, but many men belonged to more than one club. (In some cases it was socially discouraged to be a one-club man- belonging only to the Royal Automobile Club would be seen as pretty gauche, for example.)

-Clubs catered to different groups- one was for junior Army and Navy officers, another for senior officers, the Athenaeum for scholastic types, and so on. There were clubs that prided themselves on inviting members freely, but they were an exception.

-Typically, admission was on a 'blackball' invitation system. That is, when a member was proposed, each existing member was given a white marble and a black one to vote anonymously with- too many "blackballs" would end the prospective member's chances, and usually the "decent thing to do" would be to resign if you nominated someone who was rejected too soundly.

(The few that still exist tend to use a membership committee, now, instead.)

-They were not exactly social clubs, so much as a place you went to not be bothered. In most clubs, you really only spoke to people you were introduced to- nobody was there to mingle as such.

[Off-reference, here: ISTR there was one, remaining club that only began allowing talk over breakfast sometime in the 80s.]

-Axelrod's a bit vague on fees, but he did say that in the days when a club's servant might be paid 60 pounds a year, members could easily spend over 10,000 a year without blinking.

(Axelrod also recommends The Gentlemen's Clubs of London by Anthony Lejune as a reference, though I'm afraid I don't have that one myself.)

I'm not sure if you're referring to any region specifically but many of these clubs still exist and thrive in the UK. There is a prestigious district of central London called St. James's, which is known as clubland for the plurality of its Gentleman's clubs.

The clubs first began to develop in the early 18th century as fashionable establishments for the aristocracy to gamble, which was at the time illegal in Britain in public, hence the need for the clubs to be private, members-only affairs. They quickly became a pied-a-terre of choice and a place to network and mingle for the important and powerful.

There is a famous story from the early days of the notable Beefsteak Club (more below), which had caught the attention of the police for the alarming number of young men entering and leaving. They reasonably assumed it was a brothel. One evening they waited outside the club and the first man they arrested was the Prime Minister. As the second man came out a police officer said sarcastically "And I expect you're the Archbishop of Cantabury, then? To which the man replied "Well actually yes, I am..."

In general, one pays an annual fee for membership, and then for services such as overnight accommodation, food and drinks, etc.

The key point of differentiation from any other club is their exclusivity. Membership is generally restricted and members must be proposed by one or several current members and approved at committee. Each club has their own criteria, and clubs are known for appealing to people from certain industries, but in principle admittance is restricted to the upper and upper-middle classes, and even then only to people who have achieved prominence or success. Black goes into more detail in 'A Room of His Own' which explores key themes around clubland in Victorian times.

For example, the Beefsteak Club famously requires a political affiliation reminiscent of the Whig party of the United Kingdom, advocating liberty and prosperity. Notable members include the diplomat, adventurer and MP for Cumbria Rory Stewart, and the broadcaster Stephen Fry. Another is the East India Club, which requires its membership to have attended one of Britain's old public schools. Similarly, one of the oldest and most prestigious clubs is The Garrick, whose membership typically comes from the performing arts community and has included A.A.Milne and Charles Dickens in the past. Notably, the veteran broadcaster and political commentator Jeremy Paxman applied to join the Garrick and was initially blackballed (rejected in a secret ballot), highlighting the exclusivity against he who is among the most highly paid broadcasters in the UK. His own book on the British establishment, which may have contributed to his rejection, is an excellent source for further reading.