What are the rights available for a trademark under the Supplemental Register? Can I recover profits and file an injunction, should somebody infringe my trademark?
In principal, yes.
The tough part of having a trademark on the supplemental register is that you don't get the benefits of notice or presumption of validity or ownership. Basically, a registration on the supplemental register does not provide "notice" to a potential infringer, so it is more difficult to show "willfulness" in infringemnt. Also, because there is no presumption of ownership or validity with the supplemental register, if you ever end up in court, you will need to prove that the trademark is something that you own, is a valid mark, and that you can assert against the defendant 0-- you don't have to worry about these sorts of things when you are no the principal register.
Other than that, all of the other remedies (with the exception of being able to stop certain imports at the border) are available to marks on the supplemental register.