If God made man in His own image and said that it was excellent, then why are there birth defects, and isn't fixing a birth defect the same as changing your gender if you're homosexual?
For your first question, see the sermon outline "Does God Make the Handicapped?"
No, birth defects and changing genders are not the same. Being a male or female is not a defect. It is who a person is. Now, I grant you that some people are not content with who they are, just as there are some who think they are too short, too tall, or want a different color hair. But being blonde, brunette, or redhead is not a defect. If anything, wanting to change your gender is a spiritual defect because it means you are not content. "Let your conduct be without covetousness; be content with such things as you have. For He Himself has said, 'I will never leave you nor forsake you'" (Hebrews 13:5).
Birth defects cause difficulties in a person's life. Without repairs, they might not live or not live long or they might have trouble functioning in society. Repairing a birth defect improves a person. Changing what gender you appear to be is not an improvement.
A person who goes through a sex change is merely disguising who they are. If you check a person's DNA it would still say they are a man or a woman. All that changes are the outer features and those still would not function as in a real man or woman.