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

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???

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.

1) Both options are grammatically correct, but they have slightly different meanings. "She asked me if I had already done my homework" implies that the homework is expected to be completed at a specific time in the past. On the other hand, "She asked me if I had done my homework yet" suggests that the homework should have been completed by now, but there is no specific time frame. Both options are acceptable, so you can choose the one that conveys your intended meaning.

2) Yes, "after" is correct in the sentence "She said she would come the day after." It indicates that her arrival will occur on the day following the current day.

3) The sentence "While my family was abroad, I called a friend" is grammatically correct. It indicates that during the time when your family was away, you made a phone call to a friend.

4) The sentence "While my family wasn't at home, I called a friend" is also grammatically correct. It implies that when your family was not present at home, you made a phone call to a friend.

5) The sentence "When I arrived at school, the teacher was checking the class register" is grammatically correct. It indicates that the teacher was in the process of checking the class register when you arrived at school.

6) In the given context, the sentence "While I was getting ready for school, it started to rain" is grammatically correct. It implies that the rain began during the time when you were preparing for school.

The other sentences have multiple statements, so I will provide the correct options for each:

- While I was getting ready for school, my mum cooked breakfast. (Both actions happened simultaneously.)
- The toothpaste fell off the sink. (The sink is the correct word here. If it was a washbasin, you could use that term as well.)
- My mum woke up. (Assuming this happened during the time when you were getting ready for school.)
- I fell down the stairs. (In this context, "fell down" is the more commonly used phrase.)