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Yes they would. They would be handed over to the National Archives as a matter of course (except the classified ones, etc.)

It's still possible that they could be handed over from her personal account, but this bypasses the various controls that are supposed to be in place to ensure the fullest possible disclosure.

For all intents and purposes, yes, the people wouldn't know about it, but there's supposed to be some form of oversight into the actions of political officials, and this circumvents it.

If she had her own mail server, it's probably fair to assume she could have set it up as her outbound mail server and to use TLS of some sort when connecting to it.

But you could (in theory), request them under a FIOA

I don't see a reason that you would be detered from submitting the same FIOA request for these emails. I expect the likelihood of getting a response and a set of documents sent to you would be the same as it would be if her emails were hosted through the Department of State's mail server.

Because it's on a private server, not a government operated one. Gives her legal coverage.