The 4 examples really don’t meet the Tertiary definition. To move from unclassified to Tertiary “A road linking small settlements, or the local centres of a large town or city.”
While I think Unclassified may be better I usually won’t bother to change a road 1 level unless something really is out of sorts. It is why I do pay more attention to the difference between Unmaintained Track Road (not suitable for all cars and vehicles,) and unclassified (which is the level most any vehicle can drive without worry about damaging a vehicle.

Paved or unpaved really does not matter moving beyond unclassified although most higher level roads are paved that should not be a primary factor in making the decision.

Getting out of the woods is not a factor in determining if a road is a Tertiary Road, an unclassified road is a good quality road that will get you in or out.