Hot to use "that"?
 

I think these all sentences are correct. -->
1) I said that I would go with you.
2)My idea is that we will go to play football tomorow.
3)I know him whose car is pink.
4)That word means "tiger" in English.
5)I see that box that is red inside your room.
6)I saw the man [who,that] was your cousin at the airport.

I think these all sentences may be incorrect. -->
1)I said to you that that I would go with you.
2)I have an idea that we will go to play football tomorow.
3)I know him that his car is pink.
4)I know that word in English which means "tiger".
5)I see that box inside your room that is red.
6)I saw the man at the airport [who,that] was your cousin.

Do I think correctly?

Re: Hot to use "that"?
 

The problem here is that a lot of the sentences are unnatural rather than, or as well as, incorrect. Here's what we'd normally say :

1) I said that I would go with you. OK -this is correct
2) Let's go and play football tomorrow.
3)I know the man who/that owns the pink car.
4)That word means "tiger" in English. OK
5)I can see the red box (which is /that's) inside your room.
6)I saw your cousin at the airport.


1)I told you that that I would go with you.
2)I have an idea that we're going to play football tomorrow. ( = I think that we're going to ... etc)

4)I know the word in English which/that means "tiger".

Your problem here is that you're mixing up three different uses of the word "that". And they all have a completely different grammar. Check in a grammar book for :

1. That = demonstrative adjective or pronoun : Look at that car! Look at that!
2. That as relative pronoun in defining relative clauses : I read the book that you recommended.
3. That as subordinating conjunction : It's important that you remember to tell me./ He said that he would go.

I think one of the reasons you're confused about English is because you're trying to understand 101 different things all at the same time. I think you'll find it easier if you concentrate on one thing at a time.

Re: Hot to use "that"?
 

Thank you. I didn't know before that "that" can be the subordinating conjunction.

Re: Hot to use "that"?
 

Noun clauses introduced by that:
I'm sorry about that (I'm late).

Reported speech
He said that...

But
"I have an idea(?) that we(s.)'re(v.) going to play football(o.) tomorrow."
is not for Noun clauses, Reported speech,The conditional, as well as Subject or Object of a verb. So this sentence has wrong grammar. right?

Re: Hot to use "that"?
 

The correct sentence is :
I'm sorry that I'm late.
Here that is a subordinating conjunction. It joins two clauses (a clause is a sequence of subject, verb and (optionally) complement). So, the two clauses ...

Clause 1) I (S) am (V) sorry (C)
Clause 2) I (S) am (V) late (C)

can be joined by that : I'm sorry that I'm late

Other examples :

He told Mary that he couldn't go to the meeting.

Clause 1) He (S) told(V) Mary (C)
Clause 2) He (S) couldn't go (V) to the meeting (C)

It's important that you don't forget.

Clause 1) It (S) is (V) important (C)
Clause 2) You (S) don't forget (V)

I have an idea that we're going to play football tomorrow

Clause 1) I (S) have (V) an idea (C)
Clause 2) We (S) are going to play (V) football tomorrow (C)

As a conjunction, that is not obligatory in the sentence - it can be left out and is just "understood". Eg : I'm sorry I'm late; He told Mary he couldn't go etc. In general, the more informal the sentence is, the more likely that "that" will be omitted.

Meaning

that as conjunction is used when the second clause answers the question "What?" about the first. So :

He told Mary (What did he tell Mary?) ... that he couldn't go to the meeting.
It's important (What is important?) ... that you don't forget.
I'm sorry (What are you sorry for?) ... that I'm late.
I have an idea (What idea do you have?) ... that we're going to play football tomorrow.

So, it's not restricted to sentence types as you suggest. The things you mention like "reported speech" are just examples of sentences where clause 2 answers the question What? about clause 1.