No. Demeter is the daughter of Kronos and Rhea. She and Zeus are siblings.
persephone
Demeter was born from the Titan gods Rhea and Cronus, she was born a Greek goddess, and so always had her "powers".
Demeter's childhood was short lived. When she was a kid she was swallowed by her dad, Kronos. When Zeus poisoned Kronos and she came back up she was a fully grown woman.
Persephone was her main daughter; the one always spoken of in mythologies.
Poseidon had many kids. He had children with the goddess Demeter. and also i believe he had kid/kids with Aphrodite. there is more thats just what i know off the top of my head.
Ask why.
There was no "god of Demeter", Demeter was a Greek goddess.
Demeter
Demeter was not married. Demeter had two consorts in Karmanor and Iasion.
Demeter, being a harvest goddess, is often depicted carrying grains and/or or a cornucopia. She does not always have these, however.
Demeter had no sons