You're correct in that it is not linear. The damage increases sigmoidally with weight and is impacted by the number of axles.

The cost differences are huge. A primary road will have been constructed to tolerate the weight and frequency. It's much more expensive to maintain such a road.

In some cases, yes. Maintaining and improving roads will be paid out of different budgets. Federal, State, (sometimes) County, Local Municipality. Tertiary roads are almost certainly out of the local budget.

If it's not expressly prohibited, there's bound to be heavier traffic on these tertiary streets. The heavier vehicles do far more damage.

(There's also safety to be concerned with.)

It's okay to route traffic through these areas when it's an emergency. Occasional use is not going to be a significant impactor. However, with regular use, it's going to result in rapid deterioration.

If you really want to dig into this, here's a link to a study:

https://www.lrrb.org/pdf/201432.pdf

There are many, many more. If you're curious about the weight issue, it's known as ESAL and this page is pretty easy to digest:

http://www.pavementinteractive.org/loads/