First part is correct. Second part is not unless this particular aircraft has a non Jabiru wiring arrangement. On a standard Jabiru the battery can fall out of the plane and the engine will run normally till the fuel runs out.
Correct you are. I didn't go into too much detail as you generally need the battery to start it :-)
I had discounted the coils breaking down because they generally don't go U/S at the same time. I find with dodgey ones that they start to run rough about 5 minutes after takeoff (and over a certain RPM - around 2,800+ usually) and even though they are redundant, you can still notice things aren't going well when one of them is misbehaving.
The issue described smells like the stator but I'm at a loss to know why it would have stopped the engine. Doesn't sound like fuel, but does sound a little like coils, but being redundant... not sure.