And even this "simplified set", as you put it, is virtually unanswerable in my opinion. Especially for the period chosen here (staufen dynasty).
For one, the saxon nobles were an extremely unruly lot in the 12th/13th century. Come to think of it, they always were, but especially so during the Staufen era. They were rich, they were ambitious, they had been emperor before (Ottonians, Lothar v. Supplinburg), they had their own agenda, so at one point or the other they would have turned against the emperor during that period of time.

The pope is likewise hard to calculate, since we have a bright and also ambitious pope with Innocence III. Plus, you can't defeat a pope who is not weak. Attempt to do that and you have the excommunication in the mailbox, something not desirable with an unruly bunch of nobles at home (see Henry IV.).

I realize that I'm too unimaginative here. So assumed they'd have a strong position at home with all the troublemaking nobles brought to heel and a cooperative pope I'd assume that their priority no.1 would be to get a firm grip on northern italy and either literally subdue those cities or make them controllable by giving certain concessions. Expansion eastwards could probably follow, although I think it's more probable that there would have been energetic attempts to regain control of the Holy Land. If we take the Behemoth we are imagining here in control of Middle Europe and add some clever diplomacy with the Byzantines to it, it might even have been successful.

Further than that I don't dare to speculate.