Is it possible to sync EntraID with on-prem Active Directory?

Unfortunately, syncing isn't an option right now. You might want to consider using Entra Domain Services, which would give you a new Active Directory linked to your cloud setup, but you'd need to overhaul part of your infrastructure. The quickest way could be to connect your existing Domain Controller with Entra ID by matching users through soft or hard matching.

I don't think so. You’d likely need to write a PowerShell script to pull users from Azure AD and create them in your on-prem domain manually. Unfortunately, there's no way to sync passwords. If you enable sync and convert those users into hybrid users, you could face complications. My advice would be to just create users on the domain as needed and manage them separately. Plus, you’d miss out on features like MFA, so it might be better to transition your file servers to Azure storage instead.

The closest thing available would be managing groups; other syncing scenarios aren't currently supported. It really does seem like user syncing should be feasible since devices and groups can do it, but there are complexities involved. Transitioning fully to the cloud could be more realistic for larger organizations. You can check out Microsoft's resources for more.