Learn Portuguese (Portugal): Além de lavar o cabelo, nós fazemos a manicure. - Besides washing the hair, we do the manicure.

We use além de (besides / in addition to) because it connects directly to a verb (lavar) or a noun within the exact same sentence. If we split this into two separate sentences, we would use além disso (besides that / furthermore), like this: Nós lavamos o cabelo. Além disso, fazemos a manicure.

Since the subject is nós, shouldn't the first verb be the conjugated personal infinitive lavarmos?

It can be, but it is not required here. Because the subject (nós) is exactly the same for both halves of the sentence, it is perfectly natural to leave the first verb in its uninflected, dictionary form (lavar). Saying Além de lavarmos would also be grammatically correct.

Why does it say lavar o cabelo (wash the hair) instead of lavar o nosso cabelo (wash our hair)?

In Portuguese, you generally do not use possessive words (like nosso or meu) with body parts, hair, or clothing if the context already makes it obvious who they belong to. Instead, you simply use the definite article, which is o in this case.