When I run either Rayman by his Fans or Rayman Designer, the screen immediately freezes when DOSBox loads the game (so all I see is a black screen, since that is the first screen of the game, even before the Ubisoft logo).

Strangely, the sound continues to play and the game seems to respond to the keys I press; for example, when I'm in the main menu and I press Esc, I can hear the 'Are you sure you want to leave Rayman's World?' even though I can't see anything happening on the screen.

When I switch to windowed mode, the game suddenly works just fine - no frozen screen. However, the instant I switch back to fullscreen mode, the screen freezes again at the last thing visible in windowed mode (so if I'm in the main menu in windowed mode and I switch to fullscreen mode, I see a frozen main menu). Again, the game does respond to the keys I press and the sounds work fine.

Also worth mentioning is that the original Rayman game doesn't have this issue.

So at the moment, the only way I can play Rayman by his Fans or Rayman Designer is in windowed mode... but I would really like to play in fullscreen mode... what do I do?

I'm having the same problem. How did you set in windowed mode?

Try opening dosboxRayKit.conf in a text editor, and changing fulldouble to true (it's in that first list under the commented section).
That worked for me.