Drywall – Should gaps be left between sheets of drywall for any reason

Yes, knock off any bumps between coats, but there's no need to get it perfect. A screen sander on the end of a pole is the best tool for this job. And it goes without saying that you should minimize any bumps while the mud is still drying to avoid having to sand it later. Some even recommend using the premix joint compound on the last coat and, just before it dries, going over it with a slightly damp sponge to take off any bumps.

It is hard to tell from the picture but I am guessing you are using a traditional screen and you are moving the screen back and forth along one of the axes of the screen. If the less than stellar ascii art below is your screen, instead of instead of moving N-S or E-W move the screen NE-SW or NW-SE. Or start with the screen rotated 45 degrees and continue to sand N-S or E-W

+--+--+--+--+--+--+--+
|  |  |  |  |  |  |  |
+--+--+--+--+--+--+--+          N
|  |  |  |  |  |  |  |        W + E
+--+--+--+--+--+--+--+          S
|  |  |  |  |  |  |  |
+--+--+--+--+--+--+--+

A lot of times primer and paint will fill in these tiny scratches and most people will not notice even if the scratches are not filled in completely. But since it is your house and the scratches bother you go ahead and try the diagonal movements.