Firebase launched an alpha in April 2012 and got acquired in 2014, so I'm not sure how you've stopped following it for a decade because it's only existed for about that.
> 1) It used to download the whole state (write history) upon startup. Did they ever update it to send a snapshot, or a partial view of the data with lazy-loading for additional nodes?
Yes, that feature shipped in 2014.
> 2) Did they ever provide a simple push notification service? That was pretty much the only thing missing to be able to run Firebase without another server.
Firebase integrated with Google Cloud Messaging in 2016. Cloud Functions (which was also 2016-17) allowed operation without a server.
> 3) Were node aliases/shortcuts ever implemented, so a node could be a reference to another node? That would have greatly reduced the need for a SQL JOIN alternative.
No, the guidance was to duplicated data across paths (when Cloud Functions existed, folks used them).