Can I use the ValveHammer Editor to make maps for any game engine?

Probably not easily, but it does produce bsp, which is used in several games. It might be best to look into that format first. As for general compatibility and openness, J.A.C.K. might be a better choice. Modern source maps have evolved all the way from Quake, and the map editors along with them. This is why you could probably also gain some insight by looking at Quark (community there is mostly dead, but I guess it's doing OK considering its age).

Other than that, if you stick with a (gold) source mod it should definitely be possible.