Yes, the Navy, Coast Guard, and Marines (though very few)all have rescue swimmers.
No, officers cannot become rescue swimmers, enlisted only.
There are two types of rescue swimmers that I know of. One is shipboard swimmers, I'm not sure of what type of training or how to get Shipboard RSS (Rescue Swimmer School), the other is Airborne Rescue Swimmers.
Like I said, enlisted only. The first thing that you have to do is enlist in an aviation rating (AW,AT,AE,AD,etc.), Naval Aircrew Candidate School (right across from OCS), and Rescue Swimmer School. You have to pass an aviation flight physical, because you will be part of a flight crew. I went through training 10 yrs ago so the order and requirements are probably quite abit different, but here is a run down. Basic training, Naval Aircrew Candidate School, Rescue Swimmer School, AW "A" school (aviation rating), SERE school (nasty POW training), acoustics training, then the RAG (training squadron), finally the fleet. All of this training took a year and 10 months, with no breaks except for Christmas. ALL rescue swimmers are assigned to helicopter squadrons, and when I was in you had to be a Rescue Swimmer to even be a helicopter crewman.
Obviously, you need to be a strong swimmer to get into the program. The test in was something like 67 push ups, 100 situps, <11:00 run (I think), <12:00 400 meter swim (??). RSS lasted five weeks, included stuff like breaks and releases, buddy tows (400m and 800m), disentangelments (parachute in the water), bay swim, and lots and lots and lots of PT. You also learn advanced first aid proceedures and wound treatment. In order to graduate you have to complete a "multi", which is a multiple person save scenario. I had three people to save and the instructors just loved messing with the students. This is just the basics, once you complete the school you still have to be qualified by your fleet command (basically another test).
This was all a few years ago. The school had a guy drown, almost a year before I went and the school was going through a lot of changes then. I know that they had made quite a few changes the few years after I graduated, so it could be different now.
--WT