What maintenance is required after riding through deep water? The road where I live is on a flood plain and tends to flood every winter. Despite this, I have continued riding to work through the water. 

I'm trying my best to dry the bike out at the end of the day, but I have noticed now (on the third day of the flooding) that the steering has become a lot heavier and there's some rusty residue coming out from the lower race of the headset. I'm planning to clean and re-grease it tomorrow, but I'm wondering whether any of the other bearings will need the same treatment.

Specifically, I'm worried about the wheel hubs and the bottom bracket. While the water isn't deep enough to submerge them, it does mean they get a thorough soaking. Is the water likely to cause the same problem with them? What about if they did become submerged?
I had first hand experience with what a full immersion can do. Racing through 3-foot deep muddy water and diluted mud isn't as great as it sounds!

The biggest problem comes from your bearings, no matter what type they are. At the time I had a sealed square taper bottom bracket which is more or less impervious to getting grit and fluid and inside. However, repeated immersion (only over the course of an hour) caused water and dirt to get inside which caused this nasty grinding sound. Any other bearing is going to be similar. In my experience cage style headset bearings tend to dry out faster but pick up more grit, which is probably why your steering appears to be less responsive.

As for the rusty water you're seeing, it could be coming from uncoated areas within the frame, and unprotected parts going through the frame.

To answer your questions, yes you will experience similar issues. However, these could take more or less time depending on how much water they are exposed to and how well you mitigate those effects. Making sure water is removed (drying the bike, turning it on it's side or upside down) will help. You will also have an accelerated maintenance schedule which means cleaning bearings and regreasing parts. Following a proactive maintenance schedule will give your bike a much longer life than if you wait until something is wrong.