Are you trying to use a passkey to unlock the database or for authenticating with other websites? KeePass can natively do TOTP. There’s also plugins to do that. Including one that allows KeePass to be a native Windows 11 PassKey Provider.
I use KeePass + KeeAnywhere. KeeAnywhere will sync with a wide variety of cloud storage providers. Or your own S3 data bucket server (can be self hosted or on Amazon), if you prefer. Does pretty much the same thing though with versioning. Auto filling in Firefox is done with KeePassHttp-connector on the Firefox side and the KeePassHTTP plugin in KeePass. Similar to what you describe.
I use KeePass with KeeAnywhere. KeePass can natively sync over network share, FTP, or WebDav. With plugins, it can sync over SSH, FTPS, Amazon S3 compatible buckets (including open source compatible versions you host yourself), Azure, Box, Dropbox, Google Drive, OneDrive, and more.
No, KeePass. Fully open source, no cloud involved in any way, unless you want something to sync your data (the server only ever sees your encrypted database - all encryption and decryption is done locally). You can also host your own sync server using any of a variety of different protocols.
See here: https://keepass.info/help/base/placeholders.html#otp Also worth pointing out that most plugins are open source. You can read the source and compile them yourself, if you’d like.