They make a new tile backer board. It cuts like drywall. It should make your repair easier. It comes in 1/2 & 5/8” and it’s available in 4’ x 8’ sheets. It’s the same deal as densglass. Only, it’s made for tile imstallation

Yes, it will give a much better finished products. Use the self leveler and then sand before flooring installation.

When installing crown. I always shoot the crown 18” or 24” from the corner. If there’s a small gap at the bottom push the trim upwards. Opposite for gap at the top.

The easiest way to cut crown under 3-1/2”. Put the crown on the miter saw upside down from the way it should be installed. Scribe a pencil line top and bottom on the table and fence of the saw. Now, you can just cut 45° miters for inside and outside corners. Just hold your crown on those marks for the next cuts.

You can put a shim behind the hinge to make small adjustments. I usually use zip ties cut into 3” pcs. Then use painters tape to hold them in place. If you put the shim near the door stop. It will pull the door away from latch side and towards the top of the door frame. If you shim towards the barrel side of hinge. It will push the door towards the latch side and the floor. It sounds like you need to shim the top hinge at the stop. Then shim the bottom near the barrel.

The mesh is affordable and easy to install. Then just install batt insulation.