If the engine isn't getting gas, or it's lost pressure, the engine will crank over very briskly, and eventually start. It will catch gradually as fuel pressure is built up, run roughly and then smooth out. That doesn't seem to be what you're describing.

If the battery is weak, the engine will crank over slowly. Eventually, as the battery rapidly weakens further, it will stop cranking the engine over and you will here only the solenoid clattering.

The gauge on a battery charger does not load test the battery. It just tests for voltage. You can get 12 volts from a few flashlight batteries, but they won't crank your engine over. You need a load tester to check the cranking power of the battery (they are cheap and are a good tool to have). I strongly suspect your battery will fail the load test.

And yes, a weak battery can make the engine idle rough. The alternator has to work harder, with pulls on the engine more, dragging the idle down.