The clear reason was that Link Loved zelda and went to her rescue he is quite literally her knight in rusting Armor. Her Armored amour. nothing more or less
There is no confirmed reason why Link wanted to live with Zelda triangle lip. He's an adventurous, curious, selfless guy, I don't think he would be so clingy to live with the one he loves just because he loves her unless he thinks of her as a sister or mother figure. He probably went to Earth because he liked Earth, he likes how vast it is and he likes exploring there. He also wants to protect the one he loves and the triforce from monsters.