Hey there! Thanks for the post and for raising awareness about these updates. I'm happy to answer some of the questions raised here. We did have a lengthier discussion with Carolina Nymark on the #core-editor channel where we elaborated further. (Discussion linked on the PR)

Even if the menu item is hidden, the customizer can still be accessed, will the options still work?

Yes, the options will work properly but it's not guaranteed that the access to the customizer remains but whether the customizer should be accessed or not when this lands in Core is something that is yet to be discussed.

What role will the customizer have with FSE themes?

As Josepha expressed, the ultimate goal of FSE is to simplify the admin, reduce the number of concepts WordPress has to deal with.

The Site Editor and the customizer have the same capabilities but use different technologies. Having both at the same time can be confusing for users. The idea we are currently exploring is that the customizer as we know it today won't have any role for FSE themes. The site editor would be the customizer of the FSE themes.

Will it also be deprecated for non-FSE themes? How and When?

No, I believe the customizer will keep working as long as WordPress supports non-FSE themes and I don't see support for these ending anytime soon.

How do I convert existing customizer options for my updated theme?

A lot of customizer options are absorbed by Full-site editing blocks. For instance, you can edit template-parts right in your templates which is equivalent to widget areas capabilities, You can edit menus via the Navigation block, you can tweek the site description, logo via their dedicated blocks. You can also customize colors, fonts etc via the global styles panel. That said, extensibility APIs are needed in order to allow third-party global options. The current sidebar extensibility APIs and SlotFills available on the post editor are good indications of what these APIs might look like.

This is actually one of the main reasons for this change in Gutenberg 9.3. We don't really have all the answers today and as we move forward, things might change.

We'd like to make it easier for theme developers and contributors to try Full Site Editing and the site editor, help us find the gaps, run user tests and address questions like: "What options are we missing by default" and "what APIs should we provide".