What is it?
This KB article provides the basic understanding of how FogBugz/Manuscript works.
Who will use it?
How is it used?
Manuscript deliberately doesn't provide required fields or partial access to cases, instead preferring to allow users to do what might be necessary to get their work done. However, with integrations and/or customizations, it should be possible to get what you are looking for.
In the Kanban board, make comments mandatory when a ticket is moved from one Milestone to another.
When trying to enforce a certain format of case edit, e.g., requiring a comment when changing the milestone, we've found the best solution is an integration, powered by Web Hooks, that checks edits for the appropriate format or combination of fields. When a case edit that violates your workflow is found, the integration can assign the case to the person who made the edit with a comment describing the expected format.
Is it possible to limit edit access to certain fields at group level?
Is it possible to limit access in terms of milestone changes a group of users can perform?
The only built-in permissions scoping in Manuscript is by Project. Users can have no access, read, or edit all cases in a given project. A customization would make it possible for certain users to see certain fields as read-only, or limit the values available in some dropdowns, though a marginally determined user could find their way around this limitation.
How to set up email notifications when a user not belonging to a specific group perform certain milestone changes?
The best way to accomplish this would be in the form of an integration. Whenever a case edit happens that meets your criteria, send an email or whatever else you'd like to have happened.