Learn Russian: Парикмахер стрижёт меня каждый месяц. - The hairdresser cuts my hair every month.

We are learning that 'every [time period]' uses the accusative case for frequency. Why does каждый месяц look exactly like the nominative dictionary form?

Because месяц (month) is a masculine inanimate noun. For inanimate masculine nouns, the accusative case is completely identical to the nominative case. Therefore, neither the noun месяц nor the modifier каждый changes its ending here. If the time word were feminine, like неделя (week), you would clearly see the accusative case change: каждую неделю.

The English translation says 'cuts my hair', but I don't see the word for 'hair' (волосы) in the Russian sentence. Why is that?

In Russian, the verb стричь specifically means to cut hair or shear. You do not need to add the word for hair to make sense. Instead, you just make the person the direct object. So стрижёт меня literally translates to 'cuts me', which is the most natural Russian way to say 'cuts my hair'.

What is the infinitive form of стрижёт? The spelling looks tricky.

The infinitive is стричь (to cut hair). Russian verbs that end in -чь in their infinitive form often have unusual consonant changes when conjugated. For this specific verb, a -г- or -ж- appears in the stem depending on the person: я стригу (I cut), он стрижёт (he cuts), они стригут (they cut).

If my hairdresser is a woman, does the word парикмахер change to a feminine form?