Doubtful with Islam just disappearing. It depends on what happens next. If they Conquer Persia like OTL and keep moving eastward than it spreads eastward.
I have serious doubts that the Arabs of the 630s and 640s were even "Muslims" at all: certainly, they didn't call themselves that. The idea that the Islamic religion suddenly sprang out of nowhere fully-formed seems to me to be quite silly, and I think it should be rejected by historians. You'd certainly get a form of Arab monotheism that venerates Muhammad, but it'll likely have some serious differences from Islam if it's not influenced by the early years of Arab rule over the settled peoples of the Fertile Crescent.
I imagine the rest of North Africa would remain Christian as well. Could the Visigothic Kingdom survive or would it being Arian cause it's downfall still?
Fairly sure the Visigoths had converted to Chalcedonian Orthodoxy by the end of the sixth century, so that's a bit of a moot point. I think the Visigoths will do just fine for the foreseeable future.
Technically they were already losing chunks of it by the Arrival of Islam. The Vandals started a long series of destabilization of Roman rule in the region that saw increasing mobilization and automanization of the Berber communities throughout North Africa. By the time of Justinian the great city of Magnis Leptis was a shell of its former self and Byzantine control largely only fully extended along the coastline. The Visigoths kicking out the Byzantines from Hispania did not help either.
That is to say Byzantine rule in the region was largely dependent on maintaining a series of alliances and deals with the local Berber populations for nominal suzentry over the region. Piss off a important leader of one of these Romano-Berber Kingdoms and revolt was abound.
I'm pretty sure that Justinian's province was ruling after the suppression of the 540s revolt an area bigger than Roman Africa ever had been. I don't dispute your statement that the Berbers were becoming more organised and generally formidable, but I don't think that that's necessarily going to spell the doom of Roman rule over Africa. More likely, I think, is that a Syria analogue eventually develops, with the Roman province being shielded by allied Christian Berber states from hostile Berbers.
Plus there's the fact that pretty much from the moment of Justinian's conquest, resources were being focused hugely on the East for a period of warfare with the Iranians more intense that at any time since the Third Century Crisis. I think it's safe to say that the Sasanians (or whichever other dynasty takes over) are going to be a good deal quieter in the seventh century than they were in the sixth.
Could the Byzantines even hold onto any Iberian territory? It's awfully far from Constantinople, and IOTL they lost it all within a century.
It's possible, but I'm not sure why they'd want to. Like you say, it's a long way from Constantinople, and effective influence and power can be projected from holding Ceuta and the Balearics.