"El perro sintió mi mano."

Shouldn't the dog smelled my hand make more sense?

Does this mean that the dog sensed that my hand was there, that my dog felt the contact of my hand or that my dog took action to make contact with my hand?

In English, to say someone "felt my hand" can be a euphemism for saying that you hit them. When talking about a person this would be ambiguous, as they might have felt your hand in such a way as a child would to explore how it feels, or a loved one might for comfort. But we don't really think about dogs as feeling objects in this way, they simply touch them, nudge them, or do other actions that don't imply sentient reflection. So to me, this sentence sounds like "I hit the dog" :(

I thought body parts were preceded by articles: la mano. ?

"the dog smelled my hand" should be accepted too

"The dog was offended by my hand." Never touch a dog with a dirty hand.

Noted in comments, but in English the dog felt my hand would likely be interpreted as the person hit the dog with their hand..

. . . felt my hand with what? Its hand? Silly sentence.

The dog sniffed my hand