Tris seem to be run by OCD business managers too--everyone is either assigned a space in the transition area or you grab a space and that space is "yours" for the duration. The space is roughly the width of your bike plus a bit more and some area in front of or behind your bike where you can set up a small towel on the ground with your stuff laid out. OCD skills come in handy--you think through very carefully what you'll need and when you'll need it and set your transition area up so that you can access what you need quickly.

In my vast experience (ummm, two races) I've worn tri specific shorts and top and just did the whole race in that get-up, although I did put on a jersey for the bike/run portion of my first race because it was a bit chilly outside. So yes, I just get out of the water, put on my cycling shoes and helmet and gloves and get on my bike. I LOVE my tri shorts and wear them regularly for riding--the thin chamois is perfect for me. They start out wet from the swim but dry pretty quickly on the bike, although you spend a couple of miles with water running down your legs. For that reason I've learned that there's no point in carefully drying off your feet after the swim. Just get rid of the dirt/grass/sand/goose poop you picked up as best you can and get on with it. I also go with cycling sandals and no socks because that lets my feet air dry really well and I don't get blisters. There's no reason you couldn't put on regular cycling shorts after the swim, but you'll either be wearing them over swim bottoms (which I don't think would be really comfortable and would defeat the purpose of having a seamless chamois in the shorts) or you'd be mooning the world while changing from swim bottoms to cycling shorts.

Everyone has her own system and you do what works best for you. I suspect that people who do a lot of tris spend the first half dozen tweaking their system to get it just perfect for them.