How were they able to do this? Did their ships really have the ability to go out into the Atlantic without difficulty?
They did, probably in limited numbers, but the presence of Arabo-Islamic traders in the Atlantic is known since the Early Middle Ages.
I'm sure they really couldn't do this when the English controlled Ireland without seriously provoking a response.
I'm afraid you had such even in the XVIIth without real answer. While I was talking about Arabo-Andalusians, meaning the Early medieval period (when England didn't have an hold on Ireland), Barbary and Morcoccean expeditions reached up to Iceland without real repercussions.
For example, the
1631 raid in the Cork County (this book seems to focus on the matter :
The Stolen Village - Baltimore and the Barbary Pirates.) , without England being really able to do anything about it (not that it was able to do anything when places as Devon were raided as well).
Maybe when Iberia was mostly theirs, but it'd be hard to manage that kind of expedition during the peak of the reconquista.
At the peak, maybe (but giving how slaver raids were maintained late in Meditterranean and Atlantic, I would be far less certain than you there). But during the period between, roughly, 760's and 1200's? They proven be able to do so IOTL, in Mediterranea (Balearic Islands being basically the Islamic Tortuga) and while more limited in Atlantic.
What you'd need would be expeditions from Al-Andalus and Morocco more important in the Atlantic. Basically, not that much of a radical change.
Say, no Viking expansion, and it could give more room for this sort of things in the same era.
Just wondering though, does the term Saqaliba mean all European slaves or just those from the Rus and Eastern Europe? I was not aware the term being blanketed to the Iberian Slave trade as well as the one in the east.
Technically, it designate all slaves : the boarder sense was "European" slave (even if there's mention of Black Saqaliba) : Alans, Bulgarians, Greek, Frankish, Slavic, Germanic, etc. all of them were considered Saqaliba even if their religion (paganism) could be a possible part of the definition.
Another nuance could be in the geographical area : the average Saqaliba wasn't exactly the same in Iraq than Al-Andalus,
A "better" definition would be "someone coming from the country of Saqaliba"; the country of Saqaliba being itself defined as the place where slaves were taken.