To your satisfaction.
Where that mid-point is does depend on your design choices. With you I'd expect that to be at half way between the FPGA power supply rails.
Low-pass fitering the square wave output and using negative feedback to the comparator reference point to force the average it to be where you want it to be is an approach I've used (in a very different context).
Not that I've noticed.
Sure, I considsered that, but it would take 5 parts, not just two.
If you don't care to play with electronics, that's your choice.
I go to hardware meetups and meet a lot of people who need help with electronics. Last night there was a hardware meetup at Whipsaw, an industrial design outfit. They do gorgeous product designs but don't know much about electronics.
By the time the pizza showed up, there must have been about 150 people there. I've been to meets with 500.
Their facility is beautiful, every detail. But one thing that these well-funded tech outfits always have, in addition to a fabulous kitchen, is a couple of these things:
I'm not sure what these are. Some punishment device, looks like.
One meets all sorts of people at these meetups, and most need help with electronics.
John Larkin Highland Tech Glen Canyon Design Center Lunatic Fringe Electronics
LTSpice isn't designed for digital simulation. I've used tools that were, but it was a long time ago.
Whatever.
It's not a game, it's a profession.
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