Hello and welcome to Discourse!
The category list in the sidebar can be customised by clicking this icon (shown on hover) to add or remove categories as they like:
The categories can also be muted. A muted category won’t show on /categories (or at least: it will show all the way at the bottom in a hidden section).
In sidebar, yes, via the previous instruction No, sorry, only an admin can.
No search is available on that page. But you could make only a few categories and then add subcategories to them. If you then use this setting to organise the category page, it might be less long:
Hmm, not exactly. You can restrict categories to certain user groups in the security settings:
Users that can’t see the category, can’t add it anywhere. But there is no default way to let them ask for access. What would be your usecase here?
This might be of help, take a look