I have come across a couple of solutions that mostly fit this bill. Your list above is a good set of requirements for bookmark management.
Implicitly, your points throw out Google's Chrome and Microsoft's browsers because those companies have shown intense hunger for personal data. Safari's benevolent tyranny is not very extensible, so that leaves you to use Firefox, and/or a Firefox derivative like Palemoon or the Tor Browser Bundle (TBB).
1. Outertech's Linkman
From their website:
Linkman is a bookmark manager that efficiently organizes large numbers
of bookmarks. Linkman directly integrates with Google Chrome, Firefox,
Internet Explorer, Opera browsers, and supports many other browsers.
Synchronizing bookmarks between browsers and among computers is also
possible. You get the choice to manage your bookmarks by keywords
only, folders only, or folders and keywords. Linkman will extract
keywords from web sites and update moved web addresses. The bookmark
collection can be protected by a password. With a screenshot tutorial,
an introduction video, and a comprehensive PDF manual, getting started
has never been easier!
This software creates a local webserver and then there's a plugin for each browser. I've made it work with FireFox and Palemoon, but the Tor Browser Bundle seems to block part of the way this works. I didn't look into it, so it may be possible to get it to work.
On laptops/desktops, you could probably use Dropbox, SpiderOak, or Syncthing to sync the database file across multiple machines.
I have not looked in to doing this on mobile. I doubt iOS would be able to work with this, and Android would probably take a bit of work to get it going.
From their website:
With Firefox on all your devices, you can access bookmarks, tabs and
passwords with one easy sign-in.
I've done some preliminary research into how secure this is from 2nd, 3rd, and 4th parties, and they seem to had security and privacy in mind when they built it. This article goes in depth about they why and how without being too technical.
This is supposed to work on mobile too.
Their software does use their own servers/services though. If they did succeed in true Trust No One security, then that should be fine. That's a big
You do have the option to implement your own sync servers. Although a complete DIY solution also requires setting up authentication servers, which have little-to-no documentation, guides, or support. See below, emphasis mine:
Alternatively, you can also Run your own Firefox Accounts Server to
control all aspects of the system. The process for doing so is
currently very experimental and not well documented.