Aegeus : (ē′jo͞os, ē′jē-əs)
Aegeus, after whom the Aegean Sea is called, became king of Athens and father of Theseus.
Aegeus or Poseidon.
Yes.
She doesn't want Aegeus to recognize his son, Theseus, because Aegeus would then reinstate Theseus's position as heir and he would inherit the throne of Athens after Aegeus dies. Medea wants her own son, Medus, to rule Athens after Aegeus dies - she wants to remove Theseus so that this can happen.
Aegeus took so long because he needed proof that Theseus was his son.
Aegean
Aegeus, his father.
Aegeus promised Theseus that he would give her a son before he leaves. This promise does come through.
The Aegean Sea
No, he is not. He was the son of Aethra and king Aegeus.
Aegeus
With Aethra: Theseus With Medea: Medus