Your cousin is right when he says God lives in our hearts if we've given our lives to Christ -- but he's wrong when he says that this means we can't have emotional problems.
Just as Christians can have physical problems like everyone else, so we can have emotional problems like everyone else (often going back to issues when we were growing up). I often think of the prophet Elijah, who truly loved God and served Him during very difficult times. He saw God work in miraculous ways, and gained a dramatic victory over the prophets of the pagan god Ba'al (see 1 Kings 18). But later he felt he'd failed because the hearts of the people weren't changed. He sunk into a deep depression -- so deep that he told God, "I have had enough.... Take my life" (1 Kings 19:4). But God refused, and instead helped him confront the reasons for his depression and deal with them.
In a similar way, your cousin needs to face his problems and seek professional help to deal with them. I'm not a doctor or psychiatrist, but from what you say your cousin's emotional illness is real, and it's also serious. Join with others in your family to insist he gets help. If he doesn't, his situation may only get worse.
Remember: God loves us in the dark times as well as the light. Pray for your cousin, and help him realize that God loves him and wants him to be better. May Jonah's prayer become his: "In my distress I called to the Lord, and he answered me" (Jonah 2:2).