I still have the following doubts. I really hope you can help me this time, too.

Question

1) She asked: "Have you done your homework yet? "
She asked me if I had already done my homework/or had done my homework yet.
(Which of the two is grammatical?)
2) She said she would come the day after. (is “after” correct?)
3) While my family was abroad, I called a friend.
4) While my family wasn’t at home, I called a friend.
5) When I arrived at school, the teacher was checking the class register.
6) While I was getting ready for school,
it started to rain
my mum cooked breakfast
the toothpaste fell off the sink (washbasin?)
mum mum woke up
I fell off (down) the stairs, fell off or fell down???

Answer 1

1) She asked: "Have you done your homework yet? "
She asked me if I had already done my homework/or had done my homework yet.
(Which of the two is grammatical?)
Either phrasing is fine -- grammatically correct and makes sense.

2) She said she would come the day after. (is “after” correct?)
The day after what? The idea is incomplete.

3) OK

4) OK

5) OK

6) While I was getting ready for school, it started to rain.
My mum cooked breakfast.
The toothpaste fell off the bathroom counter.
(Note corrections above.)

mum mum woke up <~~??

I fell down the stairs.