Is it possible to put custom structures (or vanilla structures) in a custom dimension?

Yes, in the structure mod element, there is a field that says "Spawn world types." By default it's set to "Surface," but you can add custom dimensions to the list by clicking the green + symbol, remove selected dimensions from the list by clicking the red x symbol, or clear the entire list by clicking the red x... symbol. After saving the mod element it should only spawn in the dimensions contained within the list (if you did it correctly.)