You do realize we have an unlimited energy source burning down on us during the day? Energy isn’t the issue here. We already overproduce during the day (though electric cars might take the load off of that in the near future).
Also you don’t have to make the carbon go poof, you just have to capture it and store in in another form (that’s not in the atmosphere). So yes, if you have a way to capture it and you use solar power (or any other renewable) you can reduce the carbon in the air.
Earth isn’t a closed system.
Counter point: I know plenty of people who close the lid and then flush, then leave. So when you open the toilet you’re greeted by a floater or shit streaks over the bowl.
I flush with it open, check if it’s clean (otherwise use the brush and flush again) then leave.
If you want to close the lid you’d have to close it, flush, open it and check, clean, close it again. Are you doing that?