Learn English – Choice of tense: the story was interesting / is interesting

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".

despite is a preposition meaning not prevented by. A preposition normally attaches a noun to a sentence, for example:

He completed the marathon, despite his age.

already having paid the money is a noun phrase. We can include a verb in a noun phrase using either a gerund or an infinitive: in this case, a gerund is used. We make the present form of a gerund by putting -ing on the end of the verb- this is quite common- but we can also make the past gerund form by putting having in front of the perfect form- thus, having paid.

The main verb in the sentence is was, which is past simple. The strange bit in this sentence is the perfect form of the gerund, having paid.

There are fewer opportunities to use the past form of a gerund than the present form, and in addition, according to this NGram, its usage has declined steadily over the past 200 years.