Privacy Browser Android

I can't tell from your screenshot what you are recommending.

There're 3 buttons at the right side, what about a fourth?

Four button feels a bit busy. Also, I'm not sure I could come up with an icon that would easily indicate to a new user what that button would do.

The more I think about this feature, the more it seems to me this would be something that a person would be unlikely to change frequently. I think it might be better to put it in the settings activity.

I'm fine with this place

I actually suggest you to hardcode alphanumeric ascending order (folders first). And then remove move down & move up altogether.
Suggesting this because everytime the user creates a bookmark, then has to execute sort function. Why even bother about exposing it in UI ?

If one needs a bookmark to be placed on top, just preceed the name with dot (.) & it'll jump to the top. If vice-versa, then use tilde (~) & it'll move to bottom.

Bookmarks are always created at the bottom. Folders are always created at the top. I think that is the correct behavior, but it isn't really what this feature request is about, as enabling sorting alphabetically would override the display order.

My intention is that, when sorting alphabetically, all the folders would display above the bookmarks. That makes it slightly more complicated, but it feels like the right behavior to me.

Yep - let's keep it working this way.

As you said sorting will override custom user placed order, I believe move down/up functions are of no use. It's better to drop those functions, as users open several other bugs claiming this sort of erroneous behaviour with the existence of both sort & moving to be conflicting.
Just make sure sorting is executed whenever a new bookmark is entried. And the rest will take care of itself.

It wouldn't be a good idea to drop those functions as they are of significant use for people who do not want to sort bookmarks alphabetically. However, I do intend to hide them when alphabetical sorting is enabled, which is what I assume you meant.

Enabled in the sense the toggle would be inside settings right?

Yes, it will be inside of settings, as opposed to in the bookmark activity.