How do I mount the new work electrical boxes with the nail-in flange?
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.