Both are equally commonly used, but here's the rule you need to apply to adapt other sentences: If you leave off the redundant verb (in your example "do," which could also have been "look" again), then it sounds better to the ear if you treat the pronoun as a direct object even if it is actually a subject compliment following such verbs as be, seem, become, look and feel, among others. If you repeat the redundant verb though, then the pronoun must be treated as the subject of a clause. That sounds more complicated than it is. Really. Here are some examples using other pronouns: He eats faster than she does. OR He eats faster than she eats. (The redundant verb is stated, so the pronoun is treated as a subject; the subject form of the pronoun is used.) He eats faster than her. (The redundant verb is dropped, so the pronoun is treated as a direct object, which in this case it actually is because "eats" is an action verb.) You left earlier than we did. OR You left earlier than we left. (Redundant verb; object pronoun.) You left earlier than us. (Redundant verb dropped; object pronoun.) She seems happier than he does. OR She seems happier than he seems. (RV; OP, even though "seems" is a linking verb and doesn't actually take an object.) She seems happier than him. (RV dropped; OP) Important note: In the case of the linking verbs, the grammatically correct answer is always the subject pronoun, and in formal writing, etc., it may be required. But in speech, not many people would say "She seems happier than he" or "You look younger than I." It sounds stilted, and you might get odd looks from the majority of listeners. The same if you knock on a door and in perfectly correct English announce, "It is I." No one says that unless he's trying to be humorous, and the proper response would be something along the lines of "Come on in, I" or "Won't Your Majesty please honor us with His presence."

Yes, they are fine. They are both commonly used. You could also say 'you look younger than I' but it is old fashioned.

Yeah, they are okay. Although the second one is not usually used.