Since my last update I've sorted out the clearances I had mentioned (and a couple more that I found), and I've added copper pours / zones. I have also sorted out connecting the different grounds.
I originally intended to connect across the jumpers in the schematic after laying out the PCB. After doing that, the zones were associated with the old nets which no longer had any connections, and changing the nets to the new ground connected it to all the ground pins in the zone irrespective of which ones they should be connected to. So obviously that wasn't going to work.
After a bit of searching the internet, I found out that what I was looking for is (at least sometimes) called a "net tie". To do this in KiCad, the best way is to create a special footprint with the two pads, and a graphic line on a copper layer connecting the two – DRC ignores graphic lines, even when on copper layers. I created a footprint like that, set the jumpers' footprints to it, and it worked great.
So my next steps are going to be a bit of prototyping!