2. Sessions by definition are ephemeral. A database should not be necessary at all, an in-memory cache should suffice.
3. If you really need to distribute session data across multiple nodes, just propagate them asynchronously. Authentication and authorization are semantically idempotent operations. Having to possibly re-auth when making a cross-region request within milliseconds of logging in might be mildly annoying for the user, but consistency isn't a deal breaker here.