No, he is not. He was the son of Aethra and king Aegeus.
When Theseus' mother tells him that his real father is King Aegeus, she tells him he must take the sword and sandals that King Aegeus had buried for him back to the king and claim his birthright.
He lifted the rock and retrieved the sword and sandals that his father King Aegeus had left for him.
Aegeus wished for his son, Hippolytus, to die when he found out that Phaedra, Aegeus's wife, fell madly in love with Hippolytus. Aegeus assumed that Hippolytus loved her back, but really didn't. Aegeus got angry and sent him away. As he was sailing away, Aphrodite sent a monster from the sea and attacked him. Hippolytus was washed up on the shore, still breathing, and met Theseus there. Theseus then knew the truth, and was very upset and felt guilty. Theseus later died too.
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.
Yes.
Aegeus, after whom the Aegean Sea is called, became king of Athens and father of Theseus.
Aegean
No, he is not. He was the son of Aethra and king Aegeus.
Aegean Sea
The Aegean Sea
Aegeus
When Theseus' mother tells him that his real father is King Aegeus, she tells him he must take the sword and sandals that King Aegeus had buried for him back to the king and claim his birthright.
Aegeus, king of Athens.
Aegean Sea
it has lots of gold and vines
Athens. You answered your question already.