Excessive fret buzz on open strings can be a sign of a nut that's cut far too deep, but I sort of doubt that's the case here. Probably a different setup issue.
You're tuning down to C. In order for that to play anything like your bass would up in E, you're going to have to have some serious strings on your bass. If you're comfortable with filing the nut out a little, I would suggest putting on a light 5-string set (say, .060-0.125), minus the G. This will get your tension up, which will help prevent fret buzz and such, but it will also require a full setup. If you're not feeling up to the task, yes, take it to a luthier.
On muting, I use to always mute slightly when I played metal with a pick. It allows for much more control over the strings, which is important when you're ripping out 16ths at 200bpm. It also helps to control ringing open strings. After a while, I got in the habit of laying my pinky across the strings at all times. It helps to compensate for the lack of muting with most picking techniques.