True. But I seem to recall that there were talks about him having problem wit his testicles among his many other health problems. Then again, François II and Marie Stuart were married at a young age and François died not long after, so it could just be nasty rumors.
Maybe not entierly nasty rumors about his "constipated testicles" (while the chroniclers is hardly a model of objectivness), but the prince seems to have been quite impressionnable in conjuction of a poor wealth, and reproducing with Mary Stuart while being onto direct medical and court watch may not have helped him critically in the case of a testicle malformation (Cryptorchidis isn't always sterilous).
I'm not sure many would have pictured that catastrophic chain of events.
Well, Henri III did, hence why he decided that "f**k that" and almost litteraly beheaded the Ligue.
All will depend on how strong both factions were before Henri II's OTL death and how they will evolve under his rule with his survival.
De Guise depended more, originally, of Henri II's support, and grew more independent as Francis II let his uncle's influence grow as De Guise acted as a bicephal "Prime Minister" of sort.
ITTL I'd expect the Catholic party to be more divided with De Guise, Montmorency as possibly a good part of IOTL "malcontents", and a stronger Valois power.
Actually, I think the Queen of Scots would have likely STAYED in France had Henry II lived because she considered him to be her de facto father and it was only with his death and the early death of her husband that she had no one to keep her from Catherine de Medici's scorn.
Doing so would have definitely made her rule as Queen of Scotland meeting a quick end : Scottish nobles and Parliment were growing so tired of french tutelage that they wouldn't have accepted that.
As for Catherine de Medicis' scorn, I keep checking for this but I think it's again another part of the romanticist black legend on her : Mary Stuart was said to be ruling together France and Scotland with Francis II, and in the absence of another strong figure, could have be a problem during Charles IX's regency.
While a living Henri II wouldn't have too much issues with his son's widow (even possibly marrying her to Charles, if he really wanted to focus on Scotland), she was eventually a political problem.
Not that even if Catherine welcomed her, Mary would have remained in France. Doing so would have been, on a really short prospect, lossing her Scottish throne.
Queen of Scots's union with one of Francis's younger brothers somewhat unlikely to have been greenlighted despite what she and Henry II may have wanted.
The problem with Henry VIII's marriages didn't really concerned his first one : the pope was quite compliant removing the canonical issues, giving a dispense of virginity for Catherine.
Giving that, furthermore, the union wasn't consommated between Francis and Mary (as it was argued for Catherine) you really had less grounds to canonically prevent such an union.
The problems in such marriage would have been basically political, not canonical.