Funny you should bring this topic up. For your first question, the answer is no. Carriers have a pretty big wardroom with all sorts of officers. However, as a SWO, you can also be assigned to the carrier in a number of billet assignments. For myself, I currently have orders to the ABRAHAM LINCOLN stationed out of Everett, WA. I am slated to be the Electrical Division Officer. Another non-nuke billet (engineering mind you) is the Auxiliaries Officer.
As for your second question, I am also in the process of waiting on a response on a package I sent in not too long ago requesting nuke power training. The process of submitting the package was pretty painless. I followed the outline in the MILPERS manual and sent the package in. Pretty darn simple. The catch with me is that I didn't take the full realm of calculus and calc based physics while in college. The only reason I applied was because 1) I am an ex nuke who did pretty well going throught he enlisted program, so I know a lot of what to expect and have a pretty good fundamental base on how the nuke plant works...and 2) my XO is a nuke who is good friends with the head nuke detailer at BUPERS...so maybe a couple strings can be pulled or waivers created for the lack of my calc based physics classes.
I actually got an email from my detailer a couple days ago saying it wasn't looking good. But then again, my detailer isn't a nuke, and if I get picked up for this training, he's gonna have to find someone else to take my spot on the LINCOLN. Hopefully, in the next few days I'll have somewhat more of a better idea what their decision is. And to tell you the truth, it's not the nuke detailer's decision...it's Naval Reactors'. So, we'll see. Hope that answers your question.
Steve Wilkins