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Come on'n get your jamaharon on! There are no real rules—just don't break the weather control network.
Latnium exists
Not the federation latnium is used by non federation worlds and occasionally by fed when dealing with non fed people but nobody in the fed uses it
Im not sure they ever explicitly said it, or at least not in the Roddenberry era.
Scotty says he "bought" a boat at one point, but I've also seen long explanations that "bought" doesn't necessarily mean with money and it's just a common Human expression for when you get something.
I think it's clear that on Federation worlds you can just rock up to a replicator and get anything that isn't dangerous (and you can get the dangerous stuff with the right license/permit), but on the frontier/border areas (ie DS9) Federation citizens receive a stipend or a wage so that they can live "on the economy" along with the non-Federation citizens.
Kirk says money doesn't exist in the voyage home
While money may not exist on Earth or the core worlds (or the entire federation) per se, there still are Federation credits for use on non-federation worlds, space stations near independent powers, and for commerce with other independent nations (like the negotiations for the Barzan Wormhole). So maybe like on Ferenginar (if it can convert into Latinum) Qo'noS (if they have money), Romulus, or any other economy. So while he may not of used money (as he only bought from Federation worlds), there still is money. It also can be used for worlds which have not made the transition to a post-scarcity/money environment. And to add to that, the Bank of Bolarus implies that some money is used by the federation. So while you are right in that citizens of the federation probably get stuff for free, that doesn't mean money doesn't necessarily exist (which you did point out.)
While money may not exist on Earth or the core worlds (or the entire federation) per se, there still are Federation credits for use on non-federation worlds. So maybe like on Ferenginar (if it can convert into Latinum) Qo'noS (if they have money), Romulus, or any other economy. So while he may not of used money (as he only bought from Federation worlds), there still is money. Also the existence of the bank of Bolarus seems to imply there is a currency for the federation (credits).