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No. The private network was originally shared across all accounts, but later on they changed it to be isolated per account. It's been that way for a couple of years.

The introduction of VPC just means you can isolate within the same account.

Yes, on both Digital Ocean and its 'brother from another mother' Linode. I have a client with a few Linode VPSs and their biggest attacks by far come from the 'private' network.

Yes.

They also will automatically enable a private network interface for you if you use their Floating IP feature. This caught me by surprise when I found out the hard way :)

That suspiciously sounds like an anchor IP address and not an actual private network interface:

https://www.digitalocean.com/docs/networking/floating-ips/

Ahh you are completely correct. It caused issues for me as it added a new interface that my firewalls knew nothing about.