Jason was not a Greek god, and never became one, but in any-case yes Jason had children.
With Hypsipyle daughter of Thoas:
Euneus, king of Lemnos
Nebrophronus
With Medea, daughter of Aeetes:
Mermerus, King of Threspotia
Medus, king of the Medes
Pheres
Alcimenes
Apis
Polyxenus
Thessalus
(and others)
Among these sons was the sister of Medus, Eriopis, who with Oileus had Ajax 'the Locrian/Lesser' .
Jason is not a Greek god.
No, a mythological Greek hero.
No: Hestia the Greek goddess was a virgin.
No. Neither He nor His wife Persephone have any children.
Hera
sorry but Jason is not a god at all in fact he was the hero of the golden fleece
Myth does not claim he fathered any.
No, and technically, Thanatos is a daemon.
First: Jason was NOT a god, he was a hero. Significant difference. Second, Jason had two wives: Medea and Creusa.
Guessing you are asking what his power is, well he does not have any real power - but he can look both forwards and backwards at the same time (which is why the month January is named after him)
yes he had a son named pan who became the god of nature
Priapus and Oenopion, to name two.