How to create a page to custom display of particular node type and add it to node context menu?
First question:
First you are going to want to set your views path to: node/%node/specialized-display. Then, in the contextual filter, under "WHEN THE FILTER VALUE IS AVAILABLE OR A DEFAULT IS PROVIDED" check off: Specify validation criteria. Finally, select "Content" from the dropdown and select your node type.

Second Question:
This guide explains how to add contextual links, all you are going to need a basic custom module with a routing.yml file. You can set the route to the view. Hint, Drupal console's drupal debug:router is a great way to find the route to your view. https://www.drupal.org/docs/8/api/menu-api/providing-module-defined-contextual-links#s-rendering-contextual-links

Third question:
I've had my best luck with the sub-pathauto module to make the aliased routes. https://www.drupal.org/project/subpathauto

Thoughts...
I am using this setup on two sites so far and this works pretty well. Sub-Pathauto seems to behave fairly well. I do have one tip though, when initially building my views, I disable it, so that I can ensure everything works before I add the extra layer.

Other thoughts...
This could be done through traditional custom views processing. If you have a big site, it may be the way to go. I haven't tested Sub-Pathauto at large scale, this is currently something I am experimenting with.