There are "height limits," but whats more important are your anthros. These are measurements of distance from different points of your body to make sure you are able to fit into the various aircraft. You might be qualified for some aircraft but disqualified for others based on your dimensions. You are pretty big, and while I don't know the specific measurements, I imagine you would be cutting it close on some of them (especially ejection seat aircraft). However, there are several larger pilots on this website who I am sure will offer their advice.
A SWO starts out as basically a division officer on a ship somewhere (non-nuclear). You will lead a division of 15-40 individuals. However, this will come why qualifying for your surface warfare pin. It is a essentially a bunch of different quals that you have 18 months to complete (otherwise you fail, are transferred out of the SWO community, and an appropriate note made in your service record).
The hardest of these quals will be your Officer of the Deck qualification. Think of the OOD as the Captain's representative when he is not physically on the bridge of a ship. You will be responsible for the safe and efficient navigation of the ship, all while minding your division and working on your other quals.
I'm sure a bunch of people will pick on the SWO community in this thread, but I have a tremendous amount of respect for them. It is a hard job to come into with little experience, but you grow up fast. I have a friend in Japan right who is loving it and hating it at the same time. He gets to do some really cool things, but it is stressful because of the various demands (long hours, high expectations).
You will definitely go to some really cool places....i didn't proofread this so sorry if its just a bunch of babbling.
Btw, there are few gentlemen here who either are SWOs, lateral transferred from SWO to aviation, or earned their SWO pin while on a small boy (essentially non carrier ships) or disassociated sea tour (an aviation tour where you are at sea but not flying).
Lastly, if you are of the smarter type, I recommend seeing if you can get SWO nuke ( SWO(N) ). If accepted you will earn your SWO pin at your first ship before heading to nuke power school. After school you will serve a tour on an aircraft carrier. LOTS of money for a similar job (minus the reator stuff).