1.) If we put them in jars, they get out.

If we put them in jars, they get out. (escape)  or If we put them in jars, they get {out of the jars}.

Today I was taking out the trash and I ran {out the door}, and opened the trash can lid really fast, and a big brownish-black wasp flew off, {towards me}.

     Some students will mark "out the trash" as a prepositional phrase, but the sentence means I was taking the trash out. [Note, however, that in context "out" might also be explained as an ellipsed phrase here -- "out of the house."]
    There is no immediate answer to "Off what?" so most students will probably consider "off" to be an adverb. That is acceptable, but further thought suggests that this is an ellipsed prepositional phrase -- "off *the lid*."

Now I have to figure out how to make her.

"Figure out" = "discover"

There are several ways to explain "how to make her," but the simplest and probably the best is to consider "how" as a pronoun that functions as the direct object of "figure out." The infinitive phrase "to make her" then functions as an adjective to "how."

She wrote them down, then we chose three things for the three people to swab.

Since nothing answers the question "Down what?," students should have little trouble seeing "down" as a simple adverb.

     Technically, there is a comma-splice here, before "then." Some teachers will consider it an error; others will ignore it. The first main clause is short, and the following subject ("we") appears as the second word in the second main clause, so it is not really a problem.
     "To swab" is an infinitive. The easiest way to explain its function is to consider it as an adjective to "things." They were "things to swab for the three people." When they get into infinitives, however, some students may prefer to see "three people to swab" as an infinitive phrase that functions as the object of the preposition "for." They are things {for three people to swab}. In this perspective, "people" is the subject of the infinitive.

Then he ate them up and lay under a tree.

"Ate up" can mean "consumed," so I would not tell any student who wanted to consider "up" as part of the verb that he or she is wrong. On the other hand, "up" can be eliminated from the sentence without in any way changing the meaning. Thus "up" can also be considered an adverb here.

When he got that, she cut open the wolf and her kids came out, for, being very proud, he had swallowed them whole.

In this case, "out" can be considered an adverb, or it can be considered part of the verb ("exited"), or it can be considered a preposition in an ellipsed phrase -- "came (out of the wolf)."

     "Open" is not a preposition, but questions will arise concerning it. "Cut open" is used so frequently that the easiest thing to do here, if a question arises, is to consider "open" as part of the verb. In essence, it means she cut and she opened. Grammarians won't like that, but then grammarians get so bogged down in questions such as this that they never get students to more important questions, such as clauses. 
     On "for," See "For" and "So" as Conjunctions.
     "Proud" is a predicate adjective after "being," and "being" is a gerundive to "he."
     "Whole" is a post-positioned adjective to "them."