ranichi17 said:
What would be the effects of him not catching malaria and living long enough to marry and have issue?
It would probably delay the War of the Three Henris: things escalated pretty quickly in 1584 thanks to the Protestant Henri of Navarra becoming the heir to the Catholic Henri III, much to the chagrin of the Catholic Ligue and the Guise. As long as both Henri III and his younger brother the Duke of Anjou are alive, no one will really worry about a potential Protestant King. Note that I say delayed and not butterflied away because in my mind the situation is still pretty explosive, so I'm expecting the conflict to happen down the line.
François' marriage is likely to become an urgent matter given the fact that Henri III won't have any children (his wife is sterile and he won't divorce her) and that the Guise and the Holy Ligue aren't going to agree to Henri of Navarra becoming King (they eventually did OTL but it took Henri IV six years of war and (another) conversion to Catholicism). The question of the bride will also likely be the subject of huge and careful negotiations. Of course, if François does end up having male children, then the House of Valois won't die out or at least not immediately.
The relationship between François and Henri III was not really one of the best. François had his eyes on the throne and was quite ambitious. He also was the leader of his own political faction, the
malcontents, which was basically everyone opposing Henri III's decisions (more or less). Henri III and him only reconciliated on his deathbed OTL. That being said, since François is his brother's heir, they might still reconcile in the end if only to preserve the Monarchy and the House of Valois.
François' survival might also have an effect on the situation in the Netherlands. The Duke of Anjou was offered the title of Hereditary Sovereign of the Netherlands, but was too unpopular in the country. He also messed up when he tried to take the Netherlands by force and that ended pretty badly given that his army was slaughtered in he so-called French Fury of Antwerp in 1583. Anjou probably has no chance of getting the Netherlands but he might still dream of it.
ranichi17 said:
Would he have a rivalry with Elizabeth for breaking their engagement?
Everyone knew that Elisabeth wasn't going to marry her frog (that's an actual nickname she gave François d'Alençon): the engagement held until François messed up in Antwerp but everyone knew it was an impossible match. François could still be furious at the engagement being broken, but I'm not sure he would become a rival or an ennemy of Elizabeth. For one, the French need England's alliance against the Hapsburgs and for two, he and Elizabeth were pretty close when he came to the English court: so he might have already known the marriage wasn't going to happen. Thus, François and Elizabeth might stay on relatively good terms even after the break up of their engagement.
ranichi17 said:
What about Henri de Navarre?
He is definitely not out of the picture though his chances of getting the throne are seriously reduced. But he remains the King of Navarra, First Prince of Blood, leader of the Protestant Party, a great landowner and thus quite a powerful figure in France. He will definitely play a huge role in the events to come if the Duke of Anjou survives.