The Store does not work as good as the equivalent in Linux. Updating is often not as straightforward as it could be and sometimes not working at all. Applications with built in update routines handle it a lot better.
Plus a dash of scepticism towards trusting Microsoft with a storefront on top.
No, it’s a separate app that is completely detached from the BitWarden App and servers.
More comparable to 2Fas, Aegis or similar apps.
EDIT: the fact that TOTP is not available on the free tier always irritated me. I am happy VaultWarden has it included, though it’s only one of three places I keep my TOTPs