Can the cup get stuck or disappear inside of me?

First of all, the cup can't disappear inside you or get lost, but it can move up slightly. However, it will ultimately slip down by force of gravity.

If you have problems reaching the cup, the first thing to do is relax and breathe. If the menstrual cup has worked its way higher inside the vagina, its important to relax the muscles as tensing up will only make removing it more difficult. Try squatting and use your abdominal muscles to push the cup downwards until you can reach the stem of the cup. Pull down gently on the stem until you can reach the bottom of the cup and grasp it with your fingers and pinch it to release the suction, and gently ease it out. Remember to wait at least an hour after waking up before you try to remove your cup, as gravity often helps move the cervix and cup lower.