First sentence: I would include commas on either side of the "not..." phrase.
She heard a sound, not from the sidewalk, but from the inside of the house.
The meaning is subtly less ambiguous now and to me reads more clearly. She heard a sound, and that sound did not come from the sidewalk, it came from the inside of the house. The original sentence could also be parsed a bit less naturally as "she heard (a sound not from the sidewalk)", and the ambiguity makes it not flow quite as well.
The second sentence does remove this slight ambiguity, but takes the opposite interpretation. There was a sound from the sidewalk, and (presumably) a sound from the inside of the house, but she only heard the second one. The repetition of "sound" feels a bit strange, but without further context I don't know how I would rephrase it. (What were these sounds? Why was she expected to hear the one from the sidewalk?)
The third sentence has the same meaning as the first. She heard a sound, but it came from the house, not the sidewalk. This one also feels a bit awkward since the preposition "from" doesn't fit as naturally with the verb "to hear". If this is the structure you want, I would change the verb:
The sound came not from the sidewalk, but from the inside of the house.
"To come" and "from" fit together very naturally in English, and the verb has fewer arguments attached to it now.