Is there an overall stat cap AND is there a hard cap on single stats such as strength or intelligence? Or can I just keep gaining permanent stat upgrades from fairy chests?
There are no caps as far as I know.
In fact, the only way to progress further in NG+ mode once you have purchased all the upgrades is by farming fairy chests for stat boosts. The fairy chest stat boosts are not counted towards the castle upgrades.
Source: Beating the game on NG+5 and beyond