But I am not sure about 1. I am hesitant to say it is unacceptable.
Here in the east of England, I think we tend to use past midnight in the present tense.
A) What's the time?
B) It's past midnight.
We use it as a rough way of giving the time. It means: not long past midnight, and I guess that it is a common(ish) phrase because we are essentailly saying: the new day has begun.
However I wouldn't use it with an action verb like study, but only to suggest a state, a temporal state:
The time is past midnight. It is no longer before midnight.