De hecho, menos la primera todas son correctas. El equivalente en inglés sería así:
- ¿Estabas despierto? O te desperté... Were you awake? Or did I awoke you...
In this case it is no not correct because you're not asking, you're assuming it.
- ¿Estabas despierto o te desperté? Were you awake or did I awoke you?
This is fine. One question with a conjunction
- ¿Estabas despierto? ¿O te desperté? Were you awake? Or did I awoke you?
Fine also, there are two different questions, both of them are ok
- ¿Estabas despierto? O, ¿te desperté? Were you awake? Or, did I awoke you?
Same, two questions, in this case, the conjunction is outside the question but it is equally correct
Otro detalle a tener en cuenta, detras del interrogante "?" no se pone punto (ni coma tampoco) El interrogante cuenta como un punto, por lo tanto después siempre debes poner mayúscula