Bindings
A ski boot size 330 is about a US size 10.5-11.
It is not a shoe. It is a ski boot.
A cross country skiing boot. Along with the ski's amd poles that match.
The HEAD AdaptEdge Ski Boot is quite unique. It allows you to change the width of your boot from 104mm to 102mm with one turn of a screw.
The "last" is the width of the forefoot of the ski boot. Generally a 98mm is narrow, 102mm is medium and anything over is wide or high volume
Usually, to remove the boot from the ski, there is a lever behind the heel. When putting on the ski, it is pushed down, but it clicks upwards when the ski is attached. Thus, to remove it, push down firmly on the lever until it is in the downward position while slightly lifting your foot.
Take your boots to a ski/boot shop that sells the same brand. They can replace the heel piece, or at least order a replacement for you.
Yes...
Same as scuffs on a sneaker, scrape marks
Downhill bindings are adjustable for different size boots. Cross-country bindings only lock into the toe of the boot, so one binding works with any compatible boot.
There are various brands of ski boot warmers that are worth considering. Hotronics and Lockwoods are good makes. They can both be purchased online through Amazon.