A question ,please
How do you use and understand "do doing or do thinking"?  Could you give me some example? Thank you!

I think you are asking about the use of each single word = four in total "Do" "Doing" "Think" and "Thinking"

like the other people have clearly stated you do not put "do" in front of a verb ending with "ing"
Unless you are wishing to be very, casual and vey colloquial it would be.
"I am going to do SOME/a lot of thinking about that" 
Between the "do" and the "verb+ing" there is a quantifier "some" "a lot" "a little" "a bit"

Do = you or someone else is in the future is actively goING to engaging in something or will be in the future doing something in the future.
Doing = you or someone else is Engaging or doing something now present tense. 

In this sentence "I am going to do SOME/a lot of thinking about that" we have the verb+ing "going" but "going" will always be a FUTURE word not a present tense word.
(When followed by "to" "Going T0 ----)
(Unless it is followed by a word that TAKES the phrase into the past tense. usually this word is "going Back --- ")

It is the same for "Thinking" this can be either in the past tense or become FUTURE TENSE if associated with GOING, "going to" = Future phrase + Thinking about it /of it/over it therefore moves into the future to give Tense agreement at the begining and end of the sentence. 

This particular problem exists In English and does not In other languages. 
There is a clash and confusion between strict grammar and the individual words used in a sentence.

There is a long list of English words that naturally fit into either the past or present tense or future tense. Sometimes when there are two words one fitting more comfortably in the past and the other in the present or future, then the whole sentence takes it's priority from (usually the first word).

But they do not teach this in class rooms.

'do doing' doesn't exist as a pair of words together.
Did you mean something like, "It's going to take some doing", which means 'It's going to take some effort, or a lot of effort". 
Example: "We have a lot of work to complete before the day is over. It's going to take some doing". ie We need to do a lot of effort.

 'do thinking' only exists very informally, if at all - "I'm going to do thinking about that" ie I am going to think about it/I'm going to give it some thought. Only very rarely do I hear it. It sounds not that good and awkward (to me). You would qualify the word 'thinking' with something like 'a little', 'some' etc. , "I'm doing SOME thinking about that". "I'm doing a little thinking about it".
Example: (A) "Are you going on holidays later this year ? (B) I'm going to do some thinking about it. (I'm going to give it some thought before deciding)

Hi Jack, usually, the verbs "doing" and "thinking" don't require an additional "do" in front of them since the doing of the action is implied in the verb itself. Examples: "I was thinking of getting lunch soon." "What are you doing tonight?"