Learn Italian: Stamattina mi sono pettinato i capelli.

Because brushing your own hair is a reflexive action in Italian. In the past tense (passato prossimo), all reflexive verbs must use essere (to be) as their helping verb. That is why you use sono instead of ho.

If a woman were saying this sentence, would it change?

Yes! Because this past tense uses essere as the helping verb, the ending of the past action word (the past participle) must match the gender of the person speaking. A female speaker would change the ending and say mi sono pettinata.

Why does it say "i capelli" (the hair) instead of "i miei capelli" (my hair)?

In Italian, when you talk about actions done to your own body parts, you usually don't use possessive words like "my" or "your". The reflexive word mi (to myself) already shows whose hair it is, so you simply use the word for "the" (i).

Is "stamattina" just one word?

Yes, but it is actually a combined, shortened version of questa mattina (this morning). It is very common to hear Italians use stamattina as a single word in everyday conversation.