If you just got him a few days ago, the bits of tail that are falling off may just be the part that had fin rot. Clean, warm water is the best way to treat this, so it's possible he's starting to heal. Look for white-colored edges on the fins in a few days. That would indicate new, healthy growth.
As for your question about pain....
Several researchers published a study (in Dec 2012), saying that fish lack pain receptors. They believe that fish do NOT feel pain the way that humans do.
They compared the number of nerve fibers, which send pain signals to the brain, in both humans and fish. They found that:
- Nerves in human skin contain 83% of the specific fibers that send pain signals to the brain.
- Rainbow trout only have 5% of these fibers.
- Sharks and rays have 0% of them.
The authors say this indicates that sharks and rays probably cannot feel pain at all, and that it is "highly unlikely" for fish to feel pain.
They conclude that it was advantageous for fish to NOT feel pain, since it led to better survival in their aquatic environment.
Here is the more detailed wording, if you want to see it:
"Nociceptors are sensory receptors that respond to potentially damaging stimuli by sending nerve signals to the spinal cord and brain.....
...a typical human cutaneous nerve contains 83% C type trauma receptors (the type responsible for excruciating pain in humans)....
... rainbow trout on the other hand have only around 5% C type fibres, while sharks and rays have 0%. The absence of C type fibres indicates that signalling leading to pain perception is likely to be impossible for sharks and rays, and the low numbers (5% C fibres) suggest this is also highly unlikely for fish...
Rose et al (2013) concluded that sharks and bony fish have survived well in an evolutionary sense without the full range of nociception typical of humans or other mammals, probably because it would otherwise be disadvantageous to their survival in the aquatic environment."