Learn Italian: La mattina ci si lava e ci si pettina. - In the morning, one washes and combs one's hair.

Both lavarsi (to wash oneself) and pettinarsi (to comb one's hair) are reflexive verbs, meaning they already use the pronoun si. To say "one washes" (impersonal), we need to add another si. Because saying si si sounds awkward, Italian changes the first one to ci, creating ci si.

Since "one washes" refers to people in general, why are lava and pettina in the third-person singular?

The impersonal si (and ci si) always takes a third-person singular verb when there is no plural object following it. Since we are just saying "one washes" and "one combs" without specifying a plural object, the verbs stay singular.

Why doesn't the Italian sentence include the word for "hair"?

The reflexive verb pettinarsi literally means "to comb oneself." In Italian, this naturally implies combing your hair, so there is no need to add the word for hair (i capelli) to make sense.

Why is it just La mattina and not something with a preposition like "in"?

To say "in the morning" or "during the morning" in Italian, you typically just use the definite article with the time of day (la mattina, il pomeriggio, la sera). You don't need a preposition to translate the English "in".