NOTE: personal pronouns like who and whom have case form. The case form for who is the nominative case (often the subject of a sentence); the case form for whom is the objective case, usually a direct object.
CORRECT: I know the girl whom you are talking to.
The relative pronoun "whom" is in the objective case. This means it acts not a subject, for example, but as an object. Who is the other case, the nominative case; it's the one you would use as a subject the sentence. The rule is that the object of a preposition "to" in a prepositional phrase is in the objective case: whom.
I know the girl. You are talking to whom
CORRECT: It was the girl whom her mother punished for stealing money.
For this one, we'll remove the main clause: It was the girl. [this is the main clause; it's and independent clause, which means it can stand on it's own as a sentence.
That leaves the dependent clause: who/whom/that her mother punished for stealing money [If we take out the relative pronouns, we can see if we have a subject and a verb: her mother punished for stealing. There's the verb--punished Now where's the subject? mother. But something is missing. Who? or What? did the mother punish? The girl. The girl is the direct object [she receives the action of the verb] of the verb "punished" and so since "girl" is in the main clause, her antecedent (the word the pronoun stands for), as a direct object, it needs to be in the objective case for the personal pronoun "who" which, if you recall in the first sentence is "whom" so....
"...her mother punished whom for stealing the money."
Which, that, who Remember the relative pronoun who refers to people only; which refers to things only; that refers to either people or things.—John E. Warriner. Warriner’s English Grammar and Composition. Third Course. Liberty Edition. Orlando, Florida: Harcourt, Brace, and Jovanovich. 1986. 270. But, as you see, who strictly refers to people. It's your choice.
Alternative:
CORRECT: I know the girl that you are talking to.
CORRECT: It was the girl that her mother punished for stealing money.
You are correct about examples 1-1 through 1-4 - whom is the "correct" word, though in modern usage, who is usually used when whom would be appropriate. "Whom does he look like?" would probably sound affected or overly formal to a modern audience. As ColleenV points out, this question has been asked before, and Mechanical snail has a very good answer here: How can one differentiate between "who" and "whom"?
I also want to address examples 2-1 through 2-4. These questions all sound slightly "off". A demeanor is not a person, so you can not talk about one with the word "who". You can rephrase them as follows:
2-5- Whose demeanor is hers most like?
2-6- Whose demeanor is hers most similar to?
Hers is substituted for her demeanor. You want to know which person's demeanor is most like "hers" (whoever "she" is), so you should use the possessive pronoun whose.
That being said, I still think these "corrected" sentences sound kind of formal and awkward. You could probably get away with asking just:
2-7- Who is she most like?
2-8- Who is she most similar to?
Or to be more clear that you are talking about someone's demeanor, which I think is basically equivalent to attitude:
2-9- Who is she most like, attitude-wise?
2-10- Who is she most similar to, attitude-wise?
Best Answer
In [1] the pronoun is subject of the embedded "lives" clause, so it should be nominative "who".
In [2] the pronoun is object of "respects" so it should be accusative "whom". But nowadays people are less fussy about the distinction between "who and "whom" when the pronoun functions as object of a verb and regularly use nominative "who". In other words, both are correct in [2], but not [1].