Proof of compactness theorem from "Computability and Logic"

Question

I am having trouble understanding a proof of compactness theorem from "Computability and Logic" 5th ed. by Boolos, Burgess, and Jeffrey. I know that compactness theorem is as following

A set of first-order logic sentences has a model if and only if every finite subset of the whole set has a model.

There are some lemmas such as 'finite character lemma' and 'model existence lemma' which the book proves instead of the compactness theorem itself, and I don't get the relevance of these lemmas with the compactness theorem. I have no idea what role each lemma plays in the proof of the compactness theorem and what does it have to do with each other lemmas. My question is:

1. If a set has satisfaction properties from lemma 13.1, then does it mean that the set is satisfiable? Is it a same thing to say 'a set is satisfiable' and 'a set has satisfaction properties'?
2. Does the lemma 13.2 (finite character lemma) prooves the 'if' direction of the compactness theorem, that is, a set of sentences has a model if every finite subset of the whole set has a model?
3. Does the lemma 13.3 (model existence lemma) prooves the 'only if' direction of the compactness theorem, that is, a set of sentences has a model only if every finite subset of the whole set has a model?