Learn English – “What were you doing at 7 o’clock yesterday?” vs “What did you do at 7 o’clock yesterday?”

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.

There are two uses for using the past in your example

Did you know

Did you know that February has 29 days every four years?
Do you know that February has 29 days every four years?

The first question is more wondering if something is common knowledge for most people, the second is just a question to the person being spoken to.

P1: I didn't know how often the train comes by.
      Did you know that the train comes by every hour on the hour?
P2: Why yes I did, it's in the schedule.

It is also used to shift the timeframe of the question to the past and is meant to gain the understanding or viewpoint of the questionee at that point in time

You brought an umbrella, did you know it was going to rain?
Did you know it was going to rain before you left home?

would be asked after it has started raining, since

You brought an umbrella, do you know if it is going to rain?

would not make any sense at that point.