How to enclose and cover a well in a driveway

You could, but it strikes me as a bad idea for a few reasons. First, you'll have to replace it every year (or at least every 2-3 years). Second, you'll get much better traction on gravel, especially when wet or snowy. Finally, mulch is much more likely to wash out in the rain, especially if it's only a shallow layer.

You have divots in a concrete driveway.

I'd saw around the divot area, break it all out, and pour concrete, rather than anything to do with asphalt. Drill some holes in the edges and insert steel to tie the patch to the slab.

If following the "quick fix" fill-on-top approach, possibly thinset (tile cement) but trying to do anything with concrete in the winter is fraught with difficulty - you have to heat it up and keep it warm - if it freezes before it cures, it's toast.

My gut says that your quick fix with hot tar is going to be good for 6 months, at most. Additionally, it will complicate doing a good concrete repair later.