As I remember, the converted Saxons were Arian and some of this was political (i.e. the early church leaders had more influence within the bounds of the Roman Empire than over the Germanic tribes). From what I understand of Plontinus' post, the Muslim concept of Jesus (and by extension Muhammad) as simply a prophet or messenger but not a god himself would be in line with Arianism.
No, this is a common misconception. The Arians didn't think that Jesus was a mere man (though a great prophet). They thought that he was the incarnate Son of God. They just thought that the Son doesn't share the divine nature of the Father. So a closer parallel to ancient Arianism would be the Jehovah's Witnesses, who believe that Jesus was an incarnate archangel.
So who were the major gnostic groups? Which were the most influential?
No-one quite knows, because no-one can agree on how to categorise them or what to call them. Basically we have non-gnostic sources - notably Irenaeus, Clement, Tertullian, and a few others - who tell us what the gnostics believed and describe the various sects. Then we also have a load of texts, mostly discovered in the twentieth century, which correspond more or less to these descriptions and are therefore considered to be texts
by these same gnostic groups. But because they don't match perfectly, no-one can quite be sure which text belongs with which group, whether the traditional accounts of the different groups are accurate in the first place, or anything really.
The main groups seem to have been the Sethians and the Valentinians. This alone is tricky, though, given that the Sethians may (or may not) be identical with another bunch called the Barbelites; and the Valentinians seem to have been split into a number of small groups, notably the eastern and western wings.
This must be the first time that the first question has been judged worth its own thread.
What happened, in the end, to the 'heretical' countries outside the Roman Empire? I know that several peoples, such as the Goths and the Vandals, subscribed wholesale to the minority sects of early Christianity, but there seems to be no non-Catholic Christianity left in Western Europe by the Middle Ages. What happened?
This isn't correct. Arianism had died out long before that. JEELEN's answer is right, that the Vandal/Goth/whatever elite, although Arian themselves, found themselves ruling non-Arian populations (it's rather anachronistic and confusing to refer to the non-Arians as "Catholics", since the Arians were also Catholics - they didn't have their own church organisation as the Donatists did). Arianism had never had any success to speak of in the Roman west (the main exception being parts of Italy, notably Milan, where Ambrose had devoted great energies to out-PRing them). The barbarian rulers wisely decided not to interfere with the religious beliefs of their subjects and left them alone, and after a couple of centuries they all converted to Nicene Christianity anyway.
The reason, by the way, that the barbarians were all Arians to start with was that they were converted by Ulfilas, who spent four decades working with the Goths in the fourth century. Ulfilas was a Homoian (the main kind of Arianism), so they became Homoians too.
Note, also, that for most of the second half of the fourth century the Roman empire was officially Homoian as well, or very close to it, because this was the version of Christianity favoured by Constantius II and most of his successors (at least, the ones who cared) until Theodosius the Great. So the fact that the Goths were converting to the same kind of Christianity as the Roman emperors was diplomatically rather useful too. This also indicates what a huge challenge it was for Theodosius to reach his settlements with the Goths in the 370s, given that he was passionately anti-Arian; the fact that he achieved a lasting peace with them despite this is testament to his diplomatic ability.
Also note that the Franks don't fit into this pattern. They converted later, and they were Nicenes from the start, never Arians.
There are stories of persecutions inside the Roman Empire by Arian Kings (granted from the point of view of Catholic Roman subjects who would have reasons to exaggerate persecutions). Why wouldn't the same be true inside Barbarian territories too if the people were of a different religion than their King? I'm thinking predominantly of the Ostrogoths and the Lombards, but also ones I know very little about (e.g., Thuringians) who had prominence in the region.
I would guess that a big part of the reason was that the barbarian kings wanted to preserve Roman culture as best they could. They were impressed by it and sought political legitimacy by identifying with it. You see that both with the Ostrogoths in Italy - where they theoretically ruled in the name of the emperor - and with the Visigoths in Spain.
