Basically, What I want to do is use the tools provided in open street map, to build a road system for a game I am making, which as stated would require a custom heightmap, so is there some sort of offline version I could use for my custom heightmap?
OSM doesn’t have a height map. Third party sources are used when rendering map images with contours.
Running OSM offline is definitely possible, and is actually mandatory if you are making heavy use of it for purposes other than maintaining the map itself.
http://opengeofiction.net is one example of a fictional map created with the same tools that OSM uses. You could create something like that.
OSM’s server code is referred to as the “rails port” - see https://wiki.openstreetmap.org/wiki/The_Rails_Port and links from there for more information. Depending on what you want to do setting that up might be far more work than you need to do.
A simpler option might be to create what you need in JOSM - just don’t upload it to OSM! That’s what I did for the legend for a map style that I use - https://map.atownsend.org.uk/maps/map/map.html#zoom=17&lat=-24.987182&lon=135.17646 . The data is pretending to be in the middle of Australia, although of course it is never added to OSM; it just lives in github.