The Germanic peoples frequently traded with the Romans and picked up a lot of habits and technology from them in OTL. Realistically, they would probably become client satellite states with a hybrid culture of mostly Germanic, a good bit of Roman and bits of Slavic/Turkic/Hunnic/Magyar/Anyone else who felt like blazing a trail through Europe.
As far as other places go. Theoretically the Germans could head further down West Africa and have an economic influence on the locals at least.
Persia would probably be invaded several times over by the first Central Asian tribe to wander through but if Persia has shown us anything it is that their identity can and will survive this so I think Persia will probably become a fixture of the landscape like China. In terms of its relationship to Rome I think it will remain antagonistic, it may also become Christian, chances of the same thing happening in OTL were decent before Islam rolled into town.
India and China knew of Rome's existance through trade. We may eventually have a Roman Marco Polo come along.
I'm not going to speculate on the Americas. That's a bit too difficult and would probably require pre-established events in this ATL timeline.