Dry patch/rash in the middle of the right side of my upper lip. Entire bottom lip dry and chapped. What can this be?

Unscented lotion. Try unscented moisturizing lotion and Carmex for the lips. Skin might react more to irritants/perfumes right now. No makeup till better. Unscented laundry detergent is a good idea for her, you and the baby later!
Rash. It could be a reaction to something you ate or makeup you used. New foods, soaps, makeup are often the culprits of dermatitis. Also could be impetigo, a staph skin infection. Treatment is topical mupirocin. Less likely to be herpes which presents with fluid filled bumps after initial tingling sensation of the skin. See your doctor if it persists.
Cold sore. This may be a cold sore usually caused by a virus.