"Whilst Princess Rhaenyra misliked her stepmother Queen Alicent, she became fond and more than fond of her good-sister Lady Laena. With Driftmark and Dragonstone so close, Daemon and Laena oft visited with the princess, and her with them. Many a time they flew together on their dragons, and the princess’s she-dragon Syrax produced several clutches of eggs. In 118 AC..." -- The Rogue Prince
I think this states that Caraxes sired eggs with Syrax...or at least, Gyldayn thought so.
- 1 - Gyldayn states that Caraxes, Syrax, and Vhagar were in close proximity for extended times...after which Syrax laid eggs. Seems pretty clear they were sired by either Caraxes or Vhagar.
- 2 - Gyldayn generally considers dragons to be female if they were known to have laid eggs (as seen with the description "Vermax couldn't have been female, it was never observed laying eggs".
- 3 - Vhagar is consistently referred to as female, but Caraxes is not.
Unclear if any even hatched. Rhaenyra's 3 eldest sons had dragons by that point. The only others are Baela's Moondancer (barely larger than a hatchling), and the three hatchlings during he Dance (Stormcloud, Morghul, Shrykos).
I kind of always suspected that Caraxes and Syrax were a mated pair.
Of course, what if Gyldayn is getting his facts wrong, and the sire was Vhagar? Impossible to know, and I kind of thing GRRM would want to make this his definitive version of the facts so doubt it.