Duke keeps his eyes on his caregiver because she's holding a tasty treat just for him. If he keeps his focus on her, all of his other problems seem to fade away.
Duke expresses his love for his caregiver through his eyes. He's so thankful that she rescued him off the streets and loved him just the way he was.
Duke grows a heart on his head to express his love for his caregiver. If you see other pictures of him, you will see that it wasn't always there. He grew it just for her.