Learn English – “There was time” or “There were time”

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 agree with those in the comments, that using "What [were you]/[have you been] doing?" is much more likely to be interpreted by the recipient as rude, or at least intrusive. I can't think of a situation in which I would use either of these phrases unless I was essentially accusing someone of doing something they shouldn't have been.

If I'm interested in what someone has been doing, instead of saying "What were you doing last week?", I might say "What did you get up to last week?", although if the last time I spoke to them was much more than a week ago and there's no specific reason why last week was important, then I'd probably just say "What have you been up to?".

Disclaimer: I'm a native UK English speaker, so some of the above might be a little regional to the UK. I have spent a fair bit of time in the US and understand a decent amount of their regional variations, but I'll let a native US English speaker fill in the gaps if necessary.