I don't know if it will be sufficient to provide access to only http://files.pythonhosted.org/ and https://pypi.org/ - it's very possible that the file hosting locations will change. This is the case on a lot of other package galleries, including NuGet.org.
If ProGet can only download a portion of the files, then, there will probably be some strange errors. It's too difficult to generalize, so if you can provide us with a specific package and a specific reproduction situation, we'll be happy to try it.
But, as the error says, ProGet does only support those wheel/source packages; the legacy "egg" format (10+ years old I think) is not supported. Your developers should be able to help convert it, and you can rehost the handful of "egg" packages as needed.
As far as other errors you may see.... don't think of the "Diagnostic Center" as "checklist of things to fix", it's there to help diagnose a problem... and unless you have one reported by a user, it's probably fine. They can come from so many sources (including temporary network outages, users typing in wrong urls / passwords, etc).