It's a preposition. The words "to the playground" make up a prepositional phrase.
In this instance, to is a preposition indicating movement or an action or condition suggesting movement toward a person, place or thing.
The highlighted word "clamor" functions as a noun in the sentence. It refers to a loud and persistent noise or commotion on the playground that attracted the principal's attention.
Snow in this sentence is the verb.
The subject!
You can use the word mention in the following sentence. I wish that he would mention me in his speech.
Dog
Dangerous is an adjective.
Yes, if the sentence following the question mark and closing speech marks is the start of a new sentence.
To is a preposition in that example.
Adverb
Snow is a noun in that example.
noun