The ICW in June/July, at least in spots, will be a game of dodging powerboats and their wakes. Well, that might be anytime.
If you don't have an autopilot, it will be longish days married to the tiller, motoring or at best motorsailing. They don't call it "The Ditch" for nothing.
That said, it can be interesting. Some sections in SC and Georgia are reportedly among the most beautiful of the whole waterway. That is, if you like marsh/wilderness, birdwatching and being more or less 'alone.' I read one ICW-er's account and he just called it 'boring' and complained of seeing nothing but birds. My kind of place, but to each his own.
There are plenty of marinas along the waterway, so if you anticipate needing to stop (fuel, ice, repairs, just to get off the boat, whatever), you can. On the other hand, some stretches have good anchorages and others have limited anchoring that would get you out of the channel.
Time involved depends on how hard you want to push (how many crew to take the tiller, how many hours per day you want to run, etc. You can probably safely estimate 4-5 kt made good while underway. A lot of ICW-ers do about 10 hour days and I think many consider 50 nm a pretty darn good run for that ten hours. I would liken running the ditch somewhat similar to driving a car. How long can you drive without a break?
Morehead City is at (statute) Mile 202 and the Miami River is at Mile 1090. So, figure about 900 miles. If you do 50 miles per day (which I think will get old quick, unless you have some good relief crew to swap off tricks on the helm), that's 18 days underway. That means 10 hours per day making progress, not counting stops for fuel, waiting for bridge openings, eating out, just resting, etc.
I personally found offshore far more relaxing and enjoyable - with near ideal weather, though. Kurt and I were about 19 hours on the ICW and about 17 hours offshore. The ICW leg wore me out, and I could not sleep even when "off-watch." It was exhausting. Offshore, on the other hand, I slept like a baby, due partly to the fact that we were sailing (not motoring) and just a much more natural motion to the boat. There was less to hit offshore, too.
That said, you will have to watch the weather carefully and plan your potential inlets if you go offshore. I highly recommend downloading the relevant chapters from the Coast Pilot and studying them (Volume 4 covers the East Coast and you'll need Chapter 12 for the ICW plus whatever chapters for the offshore sections you want to do). The Coast Pilot lists ALL bridges and their vertical clearances, as well as channels and places where you might expect difficulties (confusing marks, shoaling, etc).
You can download the Coast Pilot as pdf
It goes without saying that if you plan offshore, the boat should be 'properly' prepared. Be ready for rough weather or extremely light air; good handholds below; a means to prepare meals underway; the crew is mentally prepared for an offshore run, etc, etc. If you are not used to sailing on open water, things are a bit more difficult due to the motion of the boat (moving around, etc).
As for bridges on the waterway, a lot open on a schedule, some are on-demand. During our leg from New River to Cape Fear, we had to contend with only three bridges; had we gone on to Georgetown on the ICW, there would have been an additional five. Total wait time on the three bridges was about an hour or so.
For Florida, I have no experience on the ICW, but I've read that once you hit a certain point (around Daytona??), you may as well go outside. The waterway just goes by high rise hotel after high rise hotel. There is NOTHING to see but human development.
As a final comment, I would try to plan at least one, short if need be, offshore leg just for the experience of it. For example, Georgetown to Charleston is about 40 nm and there are quite a few lit fish havens and such along the rhumb line. Except for Cape Romain, there is little hydrographical danger along the way. There would also be plenty of opportunity for offshore along the barrier islands of Georgia.
Hope this helps a little bit. That sounds like an exciting adventure, and please keep us informed! Also remember, we like pictures, lots of pictures.