In such a TL, the relationchip between, let's say, Southern France and North Africa would be much closer than the one that existed OTL after the islamic conquest of North Africa and Spain. Instead of Muslim raids in Southern France, you'd have ongoing trade in the Mediterranean sea.
The decline of Mediterranean trade predates the Islamic invasions of North Africa and Spain : it seems that byzantine gold, by exemple, stopped to come in Francia from the beggining of VII century.
While raids were destructive, they were more terrestrial than maritime before the IX century.
Without Islamic invasions, you could even have a more continuing decline of trade in Western Mediterranea (the only fleet worth of mention being the visigothic)
Culturally, North Africa and Southern Europe would be much closer than IOTL. IOTL they have many similarities, but the religious ddivide was important, and you didn't have Muslims trading in Southern France, for instance.
Yes, you had, since the VIII century.
You have early Carolingian testimony of Muslims traders in Maguelonne at the very end of the said century.
Without talking, of course, of the slave trade from North-East Carolingia towards Al-Andalus.
And that's for the earliest period : you can find arab graves in medieval harbours of Northern Mediterranean basin.
It's actually the Byzantine takeover of the Mediterranean trade that diminished the links between North Africa and Western Europe, and arguably, Islamic conquests opened it anew.
Of course, this trade is essentially a Muslim one, and European traders could really emerge in the long-range trade only from the late X century.
But it wasn't much different from Punic, Greek or Byzantine trade in this regard.
Christianity as a civilization would still have its center in the Mediterranean sea, not in Northern France, Belgium, the Netherland and the Rhinland.
I'm not that sure.
1) Christianity isn't a civilization, but managed to adapt itself to these (allowing it to not only survives, but grows).
Western Christianity, fore aforementioned reasons, can even being less focused on his southern part and I even wonder if it could emerges : OTL, Western Christianity seen as a body was essentially a Carolingian-issued concept. Before them, culturally at least, Christian realms were more distincts.
2) The economical expansion and therefore political, social, etc, of northern Europe had already began since the VIII century : long-range trade, more carrying vessels, appearance of maritime powers (Frisia, Norse petty-kingdoms,...).
Being prosperous, but less organised, it was a matter of time before they were being absorbed one way or another by their neighbor.
3) It would maybe more accurate regarding Eastern Christianity, tough. Roads such as Black Sea/Baltic seems to have been greatly influenced by Muslims realms (the number of Abassids coins in Russia is really impressive)
I found a little schema that, while really summarizing the situation, is a good explanation of what happened.
Critically, I'm not really convinced by the explanation of Carolingian fall or Vikings raids being due only due to monetary feature. It can be a partial explanation, tough.