I like to play MKDD with my own music. I get more focused and I feel I run better than I do when I play with the MKDD music. So, is it allowed to speedrun without the MKDD music/sound?

Of course, as long as that means that if you're running on emulator (or nintendon't) that the .iso isn't modded.

I personally play without using any sound, so that should be enough of an indication

Double Dash's music is awesome. Best in the series in my opinion.

If you put headphones or earpods in, you would be fine without having to change the music in your video.