the sky looks the way it does when there is going to be rain.

[kadioguy]

Take an umbrella with you. It looks like rain. [=the sky looks the way it does when there is going to be rain]

https://learnersdictionary.com/definition/rain
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Does the quoted text mean this?

It (= the sky) does (= looks) (in) the way when there is going to be rain.

1. "Does" here means "used instead of repeating a verb that has already been used".
2. "In" is optional.

Is that right?

 

[5jj]

Does the quoted text mean this?
It (= the sky) does (=looks) (in) the way when there is going to be rain. No.

2. "In" is optional. No. It is incorrect.

The sky looks the way it does when there is going to be rain.
The sky looks the way it looks when there is going to be rain.
The sky looks how it looks when there is going to be rain.

 

[kadioguy]

The sky looks the way it does when there is going to be rain.
The sky looks the way it looks when there is going to be rain.
The sky looks how it looks when there is going to be rain.

Okay, I should have said "The sky looks the way it (= the sky) does (= looks) when there is going to be rain."

"Does" here means "used instead of repeating a verb that has already been used".

As for the optional "in", please see below.

[From the Practical English Usage 3rd]

 

[5jj]

it looks in the way it does when it's going to rain is incorrect.

 

[tedmc]

Doing it (in) the wrong way.
Do it (in) any way you like.

The "in" in brackets is not optional as suggested; it is wrong.

 

[kadioguy]

it looks in the way it does when it's going to rain is incorrect.

Okay, I've got it. It is as "I don't like the way (that) you talk to me."

"The way" is an noun phrase here, which can be the object or complement of a subject. So "in" should be omitted.

Doing it (in) the wrong way.
Do it (in) any way you like.
The "in" in brackets is not optional as suggested; it is wrong.

Why? Swan says it is optional.

 

[5jj]

The "in" in brackets is not optional as suggested; it is wrong.

It is not wrong.

 

[5jj]

Okay, I've got it. It is as "I don't like the way (that) you talk to me."

You are confusing different constructions:

It looks in the way it does when it's going to rain is incorrect.
I don't like the way (that) you talk to me is possible..
I don't like the way in which you talk to me is possible..