Animals are entitled to proper care as creatures of God. But they don't have rights in the sense that people do. Men always have the right to kill an animal to keep themselves alive. Anyone who says animals have rights is denying the Christian belief that man is created in God's image. It would be literally de-humanizing to make animals our equals.
Homosexuals already have rights.
They have the same rights as any other people. But homosexual activity has no rights, because it is morally wrong. Nobody needs to accept homosexual actions, because God does not accept them. The proper place for sexual relations is in marriage. Marriage is between one man and one woman for life.