would the Fatimids be able to make themselves stronger in time to face the Mongols?
A Fatimid restoration would not be
impossible, but it would not be very likely.
And even if a powerful Caliph would arise, the Fatimids would still never become such a formidable opponent to the Mongols as the Mamluks were. The whole Mamluk government was effectively a militairy aristocracy, and I doubt a better and more dedicated army existed in the Muslim world at the time.
And then there is also another problem all Fatimid Caliphs had to deal with: they were Shi'ites, and the Shi'ites have never been the majority in Egypt, so they always had a large Sunni community and possible religious tensions to deal with, unlike the Mamluks, who were themselves Sunnites.
what was Saladin's attitude to the Assassins?
Well, the Assassins had tried to kill him several times, and it appears that they nearly succeeded in that a few times.
Consequently, Saladin organized a few campaigns againest the Assassins, although with little success, IIRC.
and could either of the Old Man(s) try to take control of Egypt, in an attempt to reunite their branch of the Shi'ite family?
just wondering.
Not quite likely, the Assassins were enemies of the Fatimids too (one of their victims was the Fatimid vizier al-Afdal)
And maybe, just maybe the Assassins would succeed in taking control of the Fatimid Caliphate,
but I don't see that happening unless the Fatimid Caliphate pretty much collapses first.
And even then, a good majority of the Egyptian Muslims at the time were Sunnites, and they weren't exactly waiting for another Shi'ite government, especially one that is dominated by the Assassins, who were commonly reviled by the other Muslims.
Still, it might not be impossible, but it is defintely one of the more unlikely scenario's.