could someone tell me what the exact rules for using forget / forget about sth are?

My observation is that forget is used with the preposition only when the verb has the meaning of "stop remembering", and not always at that.

1. I completely forgot about her birthday. (Is about optional here?)

2. I forgot my passport.

3. You are always forgetting my birthday! (In my opinion, this sounds better without the "about" but I may be wrong)

In 1. and 3., forget means "be unable to remember / dismiss for mind". In 2, it means "to leave behind".

Thanks in advance, as always.
My take on it is as follows:
If you include the word about, you are talking about the entire set of circumstances related to the object of about, not just the object itself. about sometimes gives the idea of plans regarding.

I forgot the soup. (I failed to buy the soup. I left the soup behind.)
I forgot about the soup. (I forgot my plans regarding making soup tonight.)

about can usually be omitted, because the difference is often negligible.

I forgot my passport. (I left it behind. I failed to take it with me.)
I forgot about my passport. (I forgot some event involving my passport, perhaps that business last year where it got damaged, and now I need it, and I haven't had it replaced with a new one yet.)
I forgot about my passport being damaged.

I forgot her birthday. (I failed to wish her a happy birthday. I failed to send her a birthday card.)
I forgot about her birthday. (I forgot that there was a party planned, and I failed to attend.)

The use of about can also somewhat weaken the statement, making the subject less responsible (in his own eyes).

I forgot the money. (I made a bad mistake. I failed to bring the money.)
I forgot about the money. (I was very busy, so it's understandable I would have forgotten something. That whole discussion about how much money I was supposed to bring just slipped my mind.)

One is much more likely to say (or just feel) "Oops!" with forget about.

One normally forgets things, objects, and forgets or forgets about events.

Thanks for your comprehensive and helpful answer.

P.S. I've come to the conclusion that You are always forgetting my birthday! is indeed better without about (as I suggested) because she is most probably concerned Emotion: storm with the fact that I forgot to wish her happy birthday.

EDIT: Forgot to proofread my first post. It should read "dismiss from mind".