Learn English – name of structure Being + v3

Right, as you say, when we use the past continuous tense there is usually some context that makes it clear to the reader / listener when the event was happening. Your question seems to be mostly about stative verbs (1) to be; and (2) to see.

(1) Be

Verbs that are not typically used in continuous tenses are called stative. Examples are see, be, want, hear. So we say "I was in bed..." not "I was being...". There are some cases when stative verbs like "be" are used in a continuous tense and these cases tend to change the meaning of the verb.

(2) See

As to the correctness of "I saw a very big bird at 10am this morning", this will depend on the context. It is a grammatically correct sentence; it appropriately expresses the idea that at 10am a bird appeared in view.

However, if you wanted to express some kind of continuous observation then you would need a different verb, not see (which is stative). E.g. you might say, "when someone knocked on the door at 10am I was watching a big bird".

I think the most natural way to reference the movie is:

Last night I watched a movie with a really interesting storyline.

However, if you refer to the storyline separately, in the case of a movie, it can be present or past:

Last night I watched a great movie. The storyline was/is very interesting.

The fact is that a work of literature or a film, or anything immortalized in any medium which is always accessible can be referred to in the present. However, it can also be referred to situationally in the past, in a past context.

Note that in American English, we see movies in a theater, and watch movies at home.