Last summer my wife and I stopped in at Kitty Hawk Cycle Company (http://www.kittyhawkcyclecompany.com/
) and it looked to me like they had a variety of styles of bikes to rent. Looked closer to a "real" bike shop than the other touristy rental places. We had brought our bikes, though, so didn't rent anything there.
Routes: I feel like there are some bike maps for the area, worth looking into. There are only one or two roads, depending on where you are, to choose from. I mean there are lots of roads that go East/West, but they're rather short, so your route options are rather limited. In the places where there are two North/South roads, the ocean-side road seems less busy, and the sound-side road has more businesses, so it depends on what you're looking for. There are some bike paths around Kitty Hawk, and, I think, around the section near Manteo as well.
Most sections are easily bikeable, but some are more pleasant than others. Just south of the section closest to Manteo is the Oregon Inlet Bridge. It looks to be terribly intimidating to bike on. I haven't tried it, but my friends have, and didn't enjoy it. It's a long climb to the center, and I don't think there are any kinds of bike lanes. If want to tackle that, you might plan it around high traffic areas. Personally, if I were starting in Manteo, which is about halfway up the banks, and was only going to go one direction, I'd go north. It avoids the Oregon Inlet Bridge, and the scenery is more varied. Ocracoke is a nice, little town, but the stretch of road to get to it is rather dull. I feel like for a good amount of time I was between sand dunes, close to the ocean and/or the sound, but able to see nothing but the road ahead and piles of sand on either side of me.
-- look under Regional and Local, Dare County and Pamlico for some local bike routes/facilities.