Try jumping out the ignition switch just to see if it starts. Wire from battery positive to coil positive and then crank. If it starts, then it may be either the ignition switch or the ballast. This is an easy test.
No, battery.
11.2 is nearly dead as far as batteries are rated. Perhaps during cranking the the voltage falls to 9 or 9.5.. With the starter taking all the electrons, there will be nothing left over for the coil.
Unfortunately, I don't have a spark tester. I held the number 1 wire with the plug in it on the exhaust manifold while my wife cranked the engine. No spark was visible.
