Why would

Yes, they are all correct, depending upon the context.

1. Why do you want to do that? Simple Present, with a future aspect.

You are speaking to friend who tells you that they are planning to dye their hair green. It has not yet happened, but has already been planned or decided.

You, (quite reasonably, I feel) ask the question.

" I'm planning to dye my hair green" "Why do you want to do that?"

2. Why would you want to do that?
Here, you are using a conditional, 'would', because although the plan has been made, there is still the opportunity to chage it - it is not definite.

"I might dye my hair green?" (I'm thinking about it, but I haven't decided) "Why would you want to do that?" (have you thought of changing your mind, or perhaps dying it purple instead?)

3. Why did you want to do that?

Simple past. Expresses something which happened in the past which was short lived, repeated, or longer but completed actions in the past.

"When I was younger, I wanted to dye my hair green" "Why did you want to do that? "

We would never say "Why will you want to do that." It does not make sense in English.

If we are questioning someone's definite plans for the future, we would say "Why DO you want to do that?" We use the present simple for the QUESTION because, although the plans are made, the person is thinking / talking about it NOW.

In effect the conversation is held in 2 different tenses.

"I am going to go to the cinema tomorrow" (present progressive used to talk about a plan which has already been made. I AM going to the cinema - it's definite)

"Why do you want to do that?" (present simple)

"Why would you want to do that" - this is used only informally, because it could sound as though you disagree with the person speaking. . (Why would you want to do that, I can think of something better you can do)

"Would" is used when we are talking about a future plan or idea which is NOT DEFINITE.

"I would like to buy a new car" (I'd like to, but I might not be able to afford it)

The correct question is "Would you?" This is a polite enquiry.

Why would you want to do that if you hair looks nice already? (If this is correct, why?)

We would say:

"Why would you want to do that? Your hair looks nice already" You don't need to say "if", because you are expressing an opinion. You think their hair looks nice.

This is a good example of "Why would you want to do that?" expressing disagreement. Your friend has said they want to dye their hair. Obviously they don't think it looks nice; you do - so you are disagreeing with them.

"...if you hair looked nice already? " - you are using the past tense and the future together here, and that's why it sounds odd. You need the present tense, Wyour hair looks nice"

And no.6 just isn't right for all the reasons mentioned above.

By the way, can you please tell me how you can write in bold, and underline when posting? I can't seem to do it. Thanks

This is a good example of "Why would you want to do that?" expressing disagreement. Your friend has said they want to dye their hair. Obviously they don't think it looks nice; you do - so you are disagreeing with them.

Is it incorrect to use 'if' then? If not, do the tenses have to match?
1. Why would you want to do that if you hair looks nice already? (Is this incorrect? This is an invalid conditional? If this is okay, why?)
2. Why would you want to do that if you hair looked nice already? (Correct? What does it mean compared to #1, if #1 is right?)

Thanks.