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Sync is easily achievable, I use Syncthing to go between my phone, dev server and desktop. Other options are the usual suspects, Dropbox, BT Sync, etc.

As for sharing - yeah, you probably lose that. Well, unless you sync a separate DB or something.

You can easily use it in conjunction with drop box or any other file sync program.

Yes it does you just have to run your own server.

http://keepass.info/help/v2/sync.html

it isn't difficult to share a database file for KeePass and you could easily set up a sync interface for it with github

I put my keepass in a veracrypt container and sync it over dropbox, then you can retrieve it on your phone.

That's a tad on the paranoid side since KeePass already encrypts it. If you're using the latest version which has significantly improved the encryption you should have a very good margin of security.

Its a bit of extra work to get my passwords but my entire "internet identity" is stored there so I don't mind the paranoid security.