Lemmini/Superlemmini were the two engines I used for several years before I started using NL last year. Even after a year, the former two are still my preferred engine of choice any day. While it's not exactly the authentic experience of the original game that I prefer, it's close enough, but I especially love the high resolution graphics and the smooth animation. Just be aware that the developer of SL, Tsyu, isn't as active (he comes and goes) around here as he used to be, and that some bugs have been reported to him that I would like to see fixed, most notably climber issues, as well as issues with replays, particularly with external levels.
Since you have decided to try out SL, I know this will make eric quite happy. He's been converting (in the process, he discovered plenty of bugs/issues in while converting certain levels over to SL, which I have helped him report to Tsyu and of course credited him for finding, as I never knew they existed in the SL engine) a lot of level packs to SL the last several months, which makes me quite happy since I've always wanted more packs for SL. So, I'm sure he'll be able to help you regarding the custom music. I don't know if it will work with external levels, though. As you might have seen, it plays a random music track every time you load a level file as an external level. Nevertheless, I think eric can help you with the music.
edit: Please note that any levels that require glitches to be solved will not work in SL, as SL pretty much removed all glitches. Some of your levels might work in Lemmini, though, but also note that there are plenty of glitches that are exclusive to Lemmini only and so glitch levels that work in Lemmini might not work in Lemmix, for example.