Both Unity and Unreal have "free" options, meaning if you don't make over a certain amount of money with the program, you don't have to pay for it. Both of these programs have "sample" games that you can mess with. (Since you have no interest in creating a game, I would use either's sample template to insert your music.) Of these two programs, Unreal probably is more user friendly to the non-coder, but both require at least a minimal amount of coding. Unreal uses C# while Unreal uses C++ (but also integrates a visual scripting tool called "Blueprints.")
There are two audio tools I know of for assisting in atmospheric or evolving audio/music: FMOD and Wwise. I have only ever used FMOD and if you're familiar with audio software, it will be fairly easy to pick up. I was just experimenting, but with FMOD you can do everything from program an area to play a sound effect to program a song to "evolve" as the level progresses using triggers to add the next track or the next section. FMOD would need one of the above programs to work, however. Not sure if there is a standalone software for game music, as most game music is tied to either points in the game's story or locations on the game's map.
As far as moddable games? The Fallout Series, Elder Scrolls Series, Witcher Series, Dragon Age Series, GTA...etc. They all seem to have a pretty big modding community. Check out Nexusmod or modDB for a game that interest you to mod. You will probably have to research what kind of mod kit you would need to start off, though. I have never modded a game myself.
I hope this was helpful.