Improve various capabilities checking and sanitizations

[remyperona]

Description

Add various missing capabilities checks in callbacks for admin post and admin AJAX actions, and also add or improve sanitizations where needed.

Type of change

• [x] Enhancement (non-breaking change which improves an existing functionality)

Checklist:

• [x] My code follows the style guidelines of this project
• [x] I have performed a self-review of my own code
• [x] My changes generate no new warnings
• [x] Existing unit tests pass locally with my changes