My first answer would be for you to watch the videos on time signatures and intervals again!
Time signatures are a little difficult to explain in text, perhaps we could do it on the next webinar.
As for intervals – you are spot on, it is down to how it is written, not how it sounds. This harps back to the fact that strictly speaking, D# and Eb are NOT the same note. It is only in our ‘well tempered’ system do they actually become the absolute, same pitch.
In answer to your final point, if I was going from C to Abb then it would be a ‘Diminished 6th’ interval. C to G would be a perfect 5th, but we have not been asked for that.
I suppose it’s a bit like I was told quite frequently at school & college, ‘answer the question!’
Hope this helps.