Drywall – Is it OK to liquid nail drywall in a hard-to-reach corner

I'm not 100% clear as to whether you bought the house with the cracks and the popped out nails, or whether it has happened since you bought the place. If it has happened since you bought the place then I would be concerned and have it looked into. If it was like that when you bought it the damage could have been there for decades, there's no way to know for sure unless you can speak to the former owner.

I'm assuming you've done this or you have a slab foundation, but if you have access to the foundation like a crawlspace or a basement have a look to see if there's any evidence of settling. Try and mark off the cracks and the positions of the nails in ways you can later go back and check for movement. Use a ruler to draw a pencil line perpendicular to the cracks every inch or so, and use a grease pencil for the nails, or a permanent marker. Keep an eye on them every few weeks and see what happens.

The 1/16" difference is not a problem. You are allowed up to 1/4" difference. The slight recess is much preferable to the box edge standing proud. Since nothing ever lines up precisely at this level of construction, better to err a little low than a little high.

The flange thickness is also not a problem. It will push the wallboard outwards a bit, but the resulting bulge will occur over a substantial distance, so it will not be discernible to the naked eye. It's similar to the slight bulge when the ends of wallboards are butted on ceilings and then taped over. There is an obvious bulge if you place a straight edge on it. But once everything is finished, an unaided eye cannot see it except with unusually oblique lighting.

These sort of boxes are a great solution to tight installations.