Thalia, Eucleia, Eupheme, Philophrosyne and Euthenia
Erichthonius, Cacus, and Caeculus
Yes; the names of his children are Eucleia, Euthenia, Eupheme, Philophrosyne; The Palici; The Cabeiri; Erichthonius, Periphetes, Ardalus, Cercyon (possibly), Olenus, Palaemonius (a Argonaut), Philottus, Pylius and Spinter.
Yes, in his attempted rape of Athena, the seed of Hephaestus fell to Earth (Gaea) and Erikhthonios was formed.
Hephaestus originally had one biological son, who was Erichthonius. He was 'born' when Hephaestus tried to rape Athena. However she escaped him and his semen fell to the ground, and actually impregnated Gaia. Athena became to child's foster mother. There are some myths that Hephaestus did have extra 'sons' but they themselves could be foster children Hephaestus took care of during his exile. And they were Palaimonius and the 'Kabeiroi' on the island of Samothrace - Possible a group of craftsmen, who too were slightly crippled like Hephaestus.
Hephaestus and Athena "had" one child, Erichthonius, where Hephaestus tried to go for Athena and ended up getting her leg. She wiped off her leg and threw it to the ground, where Gaia the Earth Goddess was fertilized.
Jupiter was married to Juno and they had 3 children called Hephaestus, Ares, and Hebe.
Aphrodite had no children by Hephaestus.
Hera's children were: Ares Eris Hebe Eileithyia Hephaestus All were fathered by Zeus.
Hephaestus had no "nick-names" only titles and epithets.
Hephaestus had two children by the sea nymph Cabeiro which are collectively known as the Cabeiri.
since hear is the god of marriage, she does not have any demigod children. Ares, the god of war, and Hephaestus, the god of fire are her only two godly children. it's not proper for her to have children with the mortals.
Hephaestus both was at one point a child and later had a children.
Hephaestus and Aphrodite had no childeren.
Being the goddess of marriage she could not have affairs.
Judging by how Aphrodite hated Hephaestus, I'm guessing that they didn't have any children. but that's just my guess.
Yes.
Both.