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

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 awake 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 awake you?

This is fine. One question with a conjunction

• ¿Estabas despierto? ¿O te desperté? Were you awake? Or did I awake you?

Fine also, there are two different questions, both of them are ok

• ¿Estabas despierto? O, ¿te desperté? Were you awake? Or, did I awake 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.