Are these correct?

The beginning of the sentences before "the skiers" are they still considered part of the noun phrase, or are they considered a prepositional phrase, or something different? I believe they are part of the noun phrase, but not sure.

The part in the [ ] is what I am asking about.

[Why did] the skiers crowd the slopes?
[How come] the skiers crowded the slopes?
[Which of] the skiers crowded the slopes?
[Did] the skiers crowd the slopes?
[Are] the skiers crowding the slopes?
[Don't let] the skiers crowd the slopes.
[Be careful] the skiers crowd the slopes.
[There in Colorado] the skiers crowded the slopes.
[What caused] the skiers to crowd the slopes?

Verbs never are part of noun phrases. Your sentences contain these verbs: did, come, are, do let, be careful, and what. Therefore, in those sentences the words before "the skiers" is part of the predicate.

"There in Colorado" is a prepositional phrase modifying the verb, crowded, so it's not part of a noun phrase.

The only sentence that includes the introductory words as part of the noun phrase is "[Which of] the skiers crowded the slopes?"

In all of the examples you provided, the part before "the skiers" is not part of the noun phrase. Instead, it functions as an introductory clause or a prepositional phrase. Let's break down each example:

1. [Why did] the skiers crowd the slopes?
Here, "Why did" is an introductory clause that introduces a question. It does not modify or belong to the noun phrase "the skiers".

2. [How come] the skiers crowded the slopes?
Similar to the previous example, "How come" is an introductory clause that introduces a question. It is not part of the noun phrase.

3. [Which of] the skiers crowded the slopes?
Again, "Which of" is an introductory phrase that introduces a question. It is not part of the noun phrase "the skiers".

4. [Did] the skiers crowd the slopes?
In this case, "Did" is an auxiliary verb used to form the interrogative mood of the sentence. It is not part of the noun phrase.

5. [Are] the skiers crowding the slopes?
Similar to the previous example, "Are" is an auxiliary verb used to form the interrogative mood. It is not part of the noun phrase.

6. [Don't let] the skiers crowd the slopes.
Here, "Don't let" functions as an imperative phrase, giving a command. It is not part of the noun phrase.

7. [Be careful] the skiers crowd the slopes.
Similar to the previous example, "Be careful" functions as an imperative phrase, giving a command. It is not part of the noun phrase.

8. [There in Colorado] the skiers crowded the slopes.
"There in Colorado" is a prepositional phrase indicating the location. It is not part of the noun phrase.

9. [What caused] the skiers to crowd the slopes?
"What caused" is an introductory clause that introduces a question. It is not part of the noun phrase.

To summarize, in all the examples given, the parts before "the skiers" do not belong to the noun phrase but rather serve various functions like introducing a question, forming an imperative, or providing additional information.