I just switched to Firefox and i like it. This makes it even faster: https://justthebrowser.com/ And I love the Multi-Account Containers Adon that separets cookies and other stuff.

You can try Librewolf or Waterfox. Librewolf is better on privacy ( like fingerprinting ), Waterfox quicker. And both have no Ai on board. You should use them with UBlock and I would not install to many extensions.

No. Librewolf and Waterfox let you use the Firefox Sync if you turn it on in the settings. But I don't know about any self hosting.

I use containers and made on for all google sites, one for Amazon, one for news sites, etc. And why don't you configure your Browser so that all data and caches are deleted when you close the browser? it is the same effect as private browsing without the flag private browsing. And if you want to avoid tracking, do you use a DNS service like AdguardDNS or NextDNS? And for fingerprinting you could use Mullvad or Librefox Browser. But they break some websites, both support containers.

With Librewolf or Brave you can only reopen the browser with all last pages or none. But you can ask in both browsers for each specific page you like for a exception to keep cookies and website data. This is how I'm rolling. And I block all scripts and only make exceptions when I need them on a homepage. In Librewolf I use UBlock for this, Brave has its own setting for this.