Just lovely, Hela!
I've submitted my suggestions below each of your questions. Look for this symbol =>.
1. Is “it” necessary after “watching” = "watching IT myself" ?
=> No. The ellipsis works.
2. It would be wrong to say “learn enjoying”, wouldn’t it ?
=> That's right. It would be incorrect.
3. Do we need a comma before “who” here or should the relative clause be restrictive ?
=> It's restrictive, or defining. Non-restrictive, non-defining would come across like this, "Oh, and by the way, she was a storyteller."
4. Is a comma needed after “said” ?
=> I believe the verb in 4. is "say", and although some writers don't take advantage of a comma in that context, it wouldn't hurt to add one, to signal to the reader that a new sentence is approaching.
5. same question as above
=> Same holds true for me, too.
6. Is a comma needed after “off” too? If yes, should I put it inside or outside the quotation marks?
=> A comma is not necessary there.
=> I think ... inside for BrE and outside for AmE. You should double check, though.
7. Does “as it was” mean something like “as things were”, or “given the way she behaved with us”?
=> Both. As it was [the case that she didn't tell us to get in to bed and switch the light off], we began to look forward to ...
8 & 10. How would you justify the use the progressive form here ?
=> We were expecting [at the time]
9. Does “yet another tale” mean “one more tale” here? And in such a construction should it always come the past progressive?
=> It's synonymous with still, even. (Where's your dictionary, Hela?)
=> No. For example, He's going to show us yet another book of photos!
11. Can I say that?
=> Yes. "Alas" is in my current Oxford, so apparently we're still using it these days.
12. Is “some” necessary/acceptable here or should it be omitted because we’re not talking about specific computer games?
=> It's fine, but if I were you -and I'm not - I'd change it to, "some computer game (or other)", no -s on 'game'.