The shrimp could be hiding out in your rock. I have sworn mine were gone, only to see them swaying upside down in rocky caves and coming out to take bits of fish when I fed the tank.Those aiptasia are a royal PITA. I have purchased 6 peppermint shrimp and they all have disappeared overnight. I have been using Aiptasia X to try and control it, but they keep popping up. I wish I new why the peppermint shrimp disappear.
Yup, definitely a tunicate. Feeding phytoplankton and bacteria products will help it survive. It won't harm anything, and for me, very desirable.