I would drill your holes for the rack through your cabinet, then drill some counter-sink holes on the inside of your cabinet so your bolts are flush with the inside. You can find some flat headed bolts to use for this at any hardware store. Have the head of your bolt on the inside of the cabinet and secure it with a lock washer and nut. You should not need anything too thick as far as bolts go for this. You idea on how to mount them will be fine. The bad thing is that now you will have holes in your cabinet, so if you do not like it or if you want to remove it, you have just ruined your cabinets in a sense.
The sound dampening system you describe builds a metal frame rigidly attached to clips that have rubber isolation bushings. There are two possibilities in attaching items of significant weights.
Directly to the metal framing
This would spread the load over the entire metal beam, much like attaching it to a steel stud. It would not be quite as supportive since the beams are not directly attached to each other or to a sole and top plate.
This approach would probably be sufficient for almost any load a conventional stud mounted approach might support short of things that involve dynamic pressure, such as a punching bag. As usual, mor attachment points spread out the load and reduce chance of failure. I would at least double the number of mounting screws/bolts.
However, such a mount might slightly compromise the acoustic benefits since the object would be rigidly attached to a steel member which could transmit some vibrations. Less of an issue with object that do not vibrate or emit sounds. You also need to ensure that you hit the steel beams but miss the studs and clips. If you hit the stud with a mounting screw, you defeat the clip's isolation function.
Using heavy duty drywall anchors
There are several possibilities discussed in this answer. I would tend toward the newer toggle type anchors that use a 3 inch metal bar to spread the load behind the drywall. These are probably the strongest anchors after direct stud mount. Using a toggle also isolates the load from the steel frame. Drywall is less resonant than steel, so there is less chance of compromising the acoustics, but the panel may act as a bit of a drumhead.
You might want to think of ways to free float heavy objects in front of the wall, on an easel or something similar.
Best Answer
It depends on the type of mount. If it's an articulating arm mount, the bending moment at the wall is going to be a lot higher than if you're doing a flush mount. In that case the safest option would be to open up the drywall and put 2x4s inside the metal studs, and use the included wood screws to attach the mount to the new 2x4s.
If this is a load bearing wall, and you're certain the studs are heavier 20-gauge load bearing studs, you might get by mounting an articulating arm using toggle bolts like these. If the wall is non-load-bearing, it's likely made of lighter 25-gauge studs, which just don't have the localized rigidity to support this type of mount so you're stuck reinforcing with 2x4s as I described above.
If you're doing a flush (or nearly so) mount, then you can get by with toggle bolts even if you have the lighter studs.
I wouldn't recommend any kind of self-tapping screw in a load bearing application like this. It doesn't take much overtorquing (especially with 25-ga studs) to leave them just on the verge of pulling right back out.