Answer: Yes, it is allowed to mount another filter on it, as long as the caliber is the same 

it fits in a 100 mm square filter mount, so it depends on the size of the mount adapter to the lens 

The V50x comes with a skeleton case with a slotted rear door replacement, but there is no opening to allow plugging in an external mic. As a note, the sound recording, even while in the completely waterproof case, is better than I had expected. If you're not worried about moisture, there is a threaded tripod mount on the bottom of the camera and an adapter that will allow you mount the camera without using the case. 

Yes 

1. If there is ever any more than a slight amount of rotational play, you may want to try tightening the lens mount adapter so it will hold the lens tighter to the mount. There are 3 flanges in the middle of the adapter (what the lens locks onto) that are split open slightly, this gap is what gives the tension that holds the lens tight to the body. Over time and with use these gaps decrease in size, no longer pulling the lens tight to the adapter, which allows the lens to move inside the mount more than it should.
2. To tighten the fit, get a slim metal object (a #1.5 or #2.0 small flat eyeglass screwdriver is ideal) and insert it into the slit to open it up just a little bit (see images below, make sure to do it to all 3 evenly). The more you open the gap, the tighter the lens will be to the adapter; too much and the lens will have a hard time mounting so make sure to do it just a little bit at a time to all 3 flanges evenly, it doesn’t take much at all to get a tight fit.Note: please do not use too much strength, or the shrapnel will be damaged.