I doubt if back then there was a 'professional' game. More likely part-time players. Many teams would have their ancestry back to works teams - Arsenal (the gunners) was Woolwich Arsenal - formed as a munitions workers team.
The game was initially strictly amateur and paying players was strictly against the rules. When it became obvious that the teams were making the works a profit(people were paying to go and watch them), the idea of professionalism began to take hold. Even with this being the case, paying players wasn't made legal in England until 1885 and shortly after in Scotland.
For anyone who doubts that players were getting paid unofficially, the number of Scotland internationals who headed south(Scotland almost completely dominated the first twenty years or so of football internationals) would indicate otherwise, as would the number of players who moved throughout England. Feelings were mixed though and the introduction of the leagues was seen as encouraging professionalism which was still seen in quarters as a dirty word, meaning several clubs, most notably Queen's Park, who were started as a football club and dominated the early years of Scottish football, initially refused to join the league set-up as they saw their creation in this light.
As for the history of the clubs, many of the histories are interesting.
I mean, in Scotland, you had Celtic and Hibernian being founded by the Catholic Church to help the Catholic poor, although both were professional(in the case of Hibernian, for a period illegally). You had Rangers being founded by four penniless teenage highlanders who initially couldn't even afford a football(there is a really good book about this I have read, sectarianism never reared it's head there until the 1920s). Dundee FC and Aberdeen FC were both the result of mergers so the cities could get teams in the Scottish League. Dundee United(initially Dundee Hibernian) were formed by a Liberal Councillor who wanted to create a team for the Irish community in Dundee. St. Johnstone were a cricket club who wanted a winter sport(the cricket club have since folded)! The best one is Heart of Midlothian, who in spite of their grand title, were named after a nightclub. Motherwell were formed from a merger of Works Teams. Another league team, the sadly deceased Third Lanark were formed by the British Army.
My knowledge of English football history isn't nearly as advanced, but from memory, Everton were founded as a church team and Liverpool were created because of a dispute in the Everton boardroom. Man United and Arsenal were works teams, Man City were also a church team(I suspect their founders would be rolling in their graves at the thought of the current set-up the club has).
Either way, the game was formed through people paying to watch, from ease of transport and people actually scouting talent. None of this was really possible prior to the rail network and the industrial revolution. Professional Football is a direct result of it.