Cities can use a 36 hex grid at maximum.
It seems to me that 3 is also the maximum "range" at which one can purchase new tiles. Is this correct, or is it possible to buy land further off?
No, you cannot buy tiles outside the city radius of 3 tiles. You can use a great Artist to culture bomb those tiles or a Great General to build a citadel next to those tiles if you need the strategic/luxury resources there, but don't want to build a city.
The city will continue to expand through culture up to a maximum radius of 5 tiles. But you need a lot of culture for that, the Angkor Wat wonder in that city is very useful for that.
You can never work tiles outside the 3 tiles radius, but you can get the resources on those tiles.
It is also possible to gain territory by building this a citadel with a Great General. One tactic I love is to steal some resources from the City states with that.
The downside is that it destroys the specialisation and the General on that field where you place the fortress, but I believe it's worth it, as it adds a free resource to your empire!
Also beware of the Diplomatic implications.