The two examples you gave seem grammatically correct, and they are understandable. But they also sound strange.
I would simply
I cleaned the car yesterday.
Suppose that I am your mother. Notice that in this sentence, the word the does exactly what the author mentions. It modifies the word car. This is because it should be clear to me and to you which car we are talking about.
Now suppose that yesterday, I ordered you to clean the car as punishment. Now, I ask you, "Did you do what I told you to do?" You reply
(Yes) I did the cleaning.
Notice that I did not specify what I told you to do, so you gave details by say the cleaning. Again, the word the modifies cleaning because it is clear to me and you that we are talking about cleaning the car.
If I asked, "Did you clean the car?", then you can reply
(Yes) I cleaned the car.
or you can be emphatic (extra assertive, add emphasis) and say
(Yes) I did clean the car.
Next, I understand what you mean by
I did clean the car early yesterday.
but to me, it sounds a little strange. I would suggest
I did clean the car, yesterday morning.*
I cleaned the car yesterday morning.
Also can we write as 'clean the car at early , yesterday'?
I do not think so, it sounds strange. I cannot point to any specific rule, however, to show that it is incorrect.
* There might be a slight issue with the comma. I want there to be a slight pause between car and yesterday, but a comma might not be the "correct" punctuation to use there. Maybe a dash (--).