Upvote:8
If we fly into Italy and then train or fly to other Schengen countries, will she be allowed to enter?
Can we fly out of a country like Spain etc that is Schengen as long as we enter via a country like Italy?
NO, not at all
If you don't have the required permission to enter a country, entering that country is illegal, even through Schengen, which merely abolishes border controls, and not the validity of national law of the different member states.
Italy will give her a stamp on entry, is that considered a valid Schengen visa for display if we are in other Schengen countries?
An entry stamp is just a record of you entering, this is not a visa and doesn't give you more access than you already legally have. Even if that would, an Italian border guard could never issue you a French/Spanish visa.
But you are free to go to any country you are allowed to enter visa-free (provided you don't enter any country you don't have proper permission to enter, even for transit)