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.

The business packages for AdguardDNS and NextDNS include a IP4 address. Or you try AdguardVPN, I thing 2GB per month are free. But it still can cause trouble. VPNs are not made for encrypted DNS. They are made to cover up your identity.

DoT/DoQ/DoH can cause trouble in a VPN connection and slow it down. AdguardVPN offers the option and they wrote their own protocol called TrustTunnel (it is open source since beginning of this year). I used AdguardVPN for a while and it regularly slows down the connection a lot. I activate DoH in my browser when I think I need it. Also, using a different DNS Server can be a way to help fingerprinting you. I also were keen to find a VPN with DoT/DoQ/DoH but now I'm glad Proton puts security and speed over one feature that can cause a lot of problems. And yes, they could try to put TrustTunnel in their service. But as many people always come back to the Russian connection of Adguard this will also give a backslash. There as to be a good reason why Mullvad and IVPN also don't offer to change the DNS Server to DoT/DoQ/DoH.

If you use MacOS or iOS you can try Passepartout from the App Store. You can set up a wireguard connection and also set a DNS Server with encryption. But often you manually have to change the wireguard profile to make it work. And this also with some downsides. Check the reddit forum for support.

Also, if you use a VPN it should be quite difficult to identify you if you use a non encrypted DNS server. You have to trust Proton already that they don't protocol your connection. And the DNS server only sees the Proton IP address. If you are based in Europe you can try https://dnsforge.de/ with the hard option. This includes HaGeZi Ultimate.

( I don't work for Proton and have no connection to them other than using their products. )

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.