JustARegularNerd
So you can use KeePass + Syncthing to synchronize the database file across your devices. Keeps it distributed and I’ve heard a lot of recommendations for this, although I haven’t tried it.
If you don’t want to do that, Bitwarden is well regarded and probably would suit your needs based on what you’ve said.
For my threat model, I don’t trust any online password manager, so I host my own local Bitwarden server (Vaultwarden) and use Tailscale to securely access it from any device, and if the server goes down, the Bitwarden client keeps a cached copy on the device itself.
Re the web browsers I think you’re right. You may get away with a more lightweight browser like SeaMonkey or Falkon, maybe like 1 tab of Chromium lol
Distros I’d try on that would be Linux Mint Debian Edition, Debian w/ lightweight DE like LXDE or Xfce, or Arch Linux 32 if you really want to make it minimalist. Gentoo if you’re very adventurous but with my EEE PC I found compile times took up to days.