most apartments will have somewhere to leave your bikes outside with varying degrees of cover should it rain. Your question whether to bring your bikes all depends on how long you'll be here for and whether it's worth buying a new bike when you get here. As for fixies, Tokyo has a LOT of traffic lights so I can't really see the point in riding one (come to think of it, they're just fashion accessories anyway)
Plenty of fixies in Tokyo at the moment. What bike you bring depends on what kind of riding you want to be doing. If just in the city, a fixie would be fine. If you want to do anything beyond that, I'd recommend the road bikes. There's lots of great riding to be had, but there's also a lot of hills and mountains. I've done a few touring trips - Kyoto to Tokyo, Tottori to Hiroshima, and Kyoto to Tottori by way of the San-in Coast - and they all involved climbing mountains that would've been impossible (for me at least) on a fixed gear, especially carrying a load. I don't know much about mountain biking in Japan, but at least near where I am in Tottori pretty much all of the trails I've hiked are covered with stone stairs making it not very friendly for bikes. A friend of mine brought a nice mountain bike with him but has experienced similar frustrations over places to ride it. That being said, I almost wish I had my mountain bike with me to use as a studded-tire winter rider.