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In the end game I always get problems because I have to set aside more and more cities to produce gold. At the same time production in other cities drops. Is there anything I can do? Does every city need a Library, a Temple and a Colosseum at a minimum to keep productive?

I think you might want to change the government. Monarchy is nice and everything, but IMO, later in the game, Communism(for war) and Democracy(for peace) are better. Maybe that can fix something.
Also, you're building all your cities extremely close to each other, and that reduces their maximum efficiency. Thus, you get many small cities instead of fewer bigger ones, and if you build a lot of buildings in every city, that's gonna cost you a pretty penny. I usually only have up to 5 cities at the end of the game, and that works fine.

I disagree with the last commentary. Your cities are well placed but they are not developed enough. Try more irrigations and mines, that helps a lot.
In fact, Despotism is a bummer and I usually change to Monarchy or Republic as soon as I can, that helps a lot.
Communism and Democracy are much later.

You doesn't have to use Coinage. You just need to create trade routes between your cities with caravans or freights.

Also you can build the Adam Smith' Trading Co Wonder which is going to lower your expenses.

It seems uou have many grassland tiles without roads. Roads increase trade output by +1 on many terrain types, usually from 0 to 1 (but if there's already a river; 1 -> 2) So you are missing the very basic trade income on which to build bonuses (marketplace etc only give percentage bonuses)