Are these sentences correct:

1-I woke up the man sleeping.
2-I woke up a man sleeping.

3-I woke up the sleeping man.
4-I woke up a sleeping man.

5-The man sleeping fell off his chair.
6-A man sleeping fell off his chair.

7-The sleeping man fell off his chair.
6-A sleeping man fell off his chair.
To use a present participle as an adjective it is better to put it in front of a noun. But it looks like nothing wrong with all sentences, I guess.
I wrote in my application. I sincerely request "that" you to upgrade my salary scale from present one to a more appropriate one.

But my superior ammended no need of "that"

In particular they will benefit me as I apply which i have learned to teaching mathematics to children in the school

My superior said I have learned to teach.

Can you pleasde
Students: Are you brave enough to let our tutors analyse your pronunciation?
I wrote in my application. I sincerely request "that" you to upgrade my salary scale from present one to a more appropriate one.

But my superior ammended no need of "that"

In particular they will benefit me as I apply which i have learned to teaching mathematics to children in the school

My superior said I have learned to teach.

Can you please explain,
the use of ' that ' can be omitted in a sentence.

I have learned to teach .... ( correct )

I have learned to teaching ..... ( wrong )

It is better to say " I have learned teaching mathematics to children ....... " to avoid using to many times in a sentence.
1-I woke up the man sleeping.

This is a clumsy and ambiguous sentence. Were YOU sleeping when you woke up the man?
The use of the definite article (the) implies that there were a lot of men in the room but only one was sleeping and you woke him up.

2-I woke up a man sleeping.

This is also clumsy and ambiguous. But it implies that there were many men in the room and many of them were sleeping but you woke up only one of them.

3-I woke up the sleeping man.

This is a good sentence. The use of the definite article (the) implies that there was only one man who was sleeping and you woke him up.

4-I woke up a sleeping man.

This is a good sentence, but it implies that there were many men in the room and many of them were sleeping but you woke up only one of them.

5-The man sleeping fell off his chair.

This is a clumsy sentence. However, it might be better if you said, "The man who was sleeping fell off his chair."

6-A man sleeping fell off his chair.

This is a clumsy sentence. It might be better if you said, "A man who was sleeping fell off his chair."

7-The sleeping man fell off his chair.

This is a good sentence.

6-A sleeping man fell off his chair.

This is also a good sentence.

(I hope these sleeping men did not hurt themselves when they fell off their chairs!)
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So ,you mean you woke up the sleeping man and then sleeping man fell off his chair. Why did you weak him up in the first place? He might have been having wonderful dreams. Is this sentece correct? I mean the last sentence.