I've noticed that the directory ~/Library/Safari/Icons tends to fill up
my home directory and eventually I run out of space.

I already suppress browser caching by creating a file in
~/Library/Caches/ called "Safari" with permissions 0.

What goes into the Icons folder, why is it eating my hard drive, and is
it advisable to try to prevent Safari from writing to it?

I'm on a broadband connection and my understanding it that it's
generally as fast or faster to go out to the network rather than to
cache files on the hard drive.