"I am looking at my nails."
Translation:Je regarde mes ongles.
It's true that possessive pronouns are usually avoided with body parts. In fact, for sentences structured like this it's usually replaced by a indirect object pronoun and a definite article:
I wash my hands = "Je me lave les mains" rather than "Je lave mes mains"
I shake his/her hand = "Je lui serre la main" rather than "je serre sa main"
However, when it comes to the verb regarder, the expected sentence "Je me regarde les ongles" sounds unnatural for some reason, and "je regarde mes ongles" is used instead. But you should consider it an exception.
Actually, Je me regarde les ongles is completely natural. It just sounds overly pedantic and sophisticated.
And this is true for all what you've mentioned: Je lave mes mains and Je serre sa main are completely okay, but a bit too simple. In fact, when not using an indirect object pronoun, but a possessive adjective instead, you create some distance between the body parts and the person. Saying something like Je lave mes mains sounds as if you cut your hands and now you're washing them. On the other hand, saying Je me regarde les ongles sounds really narcissistic as if you are admiring your nails as part of yourself.