Is "she's" an abbreviation of "she has"?

Is this expression "is that unheard of?" a common one?

"Is that unheard of" is not a common expression, however, the phrase "unheard of" is common. If something is "unheard of", it is new, unprecedented, strange, or unfamiliar. Ex: Eating dessert before dinner is unheard of.

Why's he listing all those terms here?

Because he has memorized all of the questions and their order so he lists all of the answers once after another without letting Mrs. Puff ask the questions. It is implied that he has done the oral test many times before and has memorized the questions from experience.

Does "spittin' image of someone" means "look alike someone"? and is it general expression?

Yes, the "spittin' image", or "spitting image", of someone means that this person looks exactly like someone else. It is a general expression and can be used for people or things, but is most commonly used when referring to people, especially in the case where a child resembles their parent. Ex: His children are the spitting image of him. Ex: The style of the home is the spitting image of an Italian villa.

In this case, does "we" mean "Sheldon" and "Mary" or "Mary" and "Penny"?

"We" is referring to Sheldon and Amy. When using we in this way it generally refers to the speaker and the person the speaker is speaking to.

What's "made it to" mean?

If you "made it to" somewhere or something, it means that you arrived somewhere or reached the goal. "Made" is the past tense and past participle. "Make" is the base form of the verb. Ex: I hope to make it to Paris by next year. Ex: She made it to the end of the book.