the tiafas had managed to consolidate around Seville?
The problem there is that even if taifas tended to unite by the mix-XIth century into larger entities, they still fell under Christian dominance at various degrees, which drained their legitimacy (which was partially based on the fight against Christians) and their ressources due to tributes.
The strategical weaknesses of taifas were the same than al-Andalus during Umayyads, as in an extreme dependence on mercenaryship or recruitment from Christian Spain and Maghrib. While Umayyad Spain could, trough clientelism and military force, control the immediate geopolitical neighbourhood, there wasn't one taifa really able to do so, meaning the various Berber tribes were free to undergo their traditional cycle of unification/split/unification whom obvious targets were Ifriqiya and al-Andalus, further destabilizing these regions (and in the case of Spain, at the benefit of Christians).
A total unification around Seville, you might say, may be the solution but I find it unlikely : as said above, the various taifa did underwent regional unification but they reached an equilibrium that couldn't be easily overlooked on risk of overextension : not only logistical, but as well from legitimacy (again, the legitimacy came from a fight against Christians, which nobody was able to pull, and dynastico-religious which none of local dynasties could claim)
What are the medium and longer term prospects of this new Al-Andalus, or by contrast, of the Reconquista?
If, by some happenance, you don't have Berbers managing to form an unified super-chiefdom as it regularily happened from this point onwards (you see I'm not exactly enthusiastic about chances), you'd end up with a quicker Reconquista. Taifa were really weakened by raids, tributes and even inner settlement (which was a thing since late Umayyad period), to say nothing of ethnic/religious tensions, and were a prey for Christian kingdoms, which would pull the same thing they did to the Taifa of Toledo, for exemple.
Altough I could see other Christians joining the party : Normans and Meridionals did IOTL joined their forces with Christian princes, and the prospect of more glory, wealth and so on might provide even more of them.
More generally, how does this change the culture, demographics (religious, racial, linguistic, etc), and what not of the Iberian peninsula?
Maybe a more archaic take on Spanish, less because absorbating Mozarabic dialects, than a lesser Basque influence on language.
You might see Portugal being butterflied as a separate political entity, altough I wouldn't write off a possible maintained Galaico-Portugese continuum not just on language but on politics.
You'd certainly have Christians to deal with a more mixed Andalusia than IOTL : Christian and Jewish communauties were still pretty much significant in these (altough still mainly Islamic by the late XIth) which wouldn't be possible to just get rid of as they did for Jews in 1492, or to quickly assimilate (if imperfectly) based on territory as for Moriscos.
It would be likely to see, IMO, a mix between how Latin States were managed as in general disinterest on systematical conversion, strategical settlement policy (as IOTL, essentially from southern France altough I could see, as said above, a Norman* settlement as well ITTL) which would not only have an effect on linguistic (greater Occitano-Romance ensemble?) but as well on political conceptions (making Spanish medieval institutions closer to French-influenced Portuguese and Catalans' ones).
*Let's be clear : by Normans I mean not just people from Normandy, but the usual mix of Normans, Angevins, Bretons and surrounding French groups you had in Italy or England.
And given these general changes, what can we infer about how history is changed?
It would look a lot like southern Italy, meaning not only a certain drive to raid and control surrounding shores (there Maghrib), and with a strong plantation economy as in Sicily, altough bigger.
Its big advantage over Sicily as it happened IOTL would certainly be a closer royal power, and less neglect and predatory economy from it.