Fill in the blank with either 'do' or 'does': "Costa Rica, one of the few countries in the world that ___ not have an army, educates its people...

You are right that this is a question about subject/verb agreement, but we are dealing with multiple clauses. For our purposes, we are only interested in the appositive clause, "one of the few countries in the world that does not have an army." For the purpose of our question, we can ignore the clause about Costa Rica educating its people.

Subject/verb agreement is tricky sometimes if there are prepositional phrases, so let's take them out too. "Of the few countries" and "in the world" are prepositional phrases, so "countries" and "world" cannot be our subject. That leaves the "one" to be the subject of this phrase, which is obviously singular and takes the verb "does."

Similarly, in your other example, we can ignore the prepositional phrases "of the few girls" and "in the school". This leaves us with the sentence, "I am one who is in 11th grade." On the other hand, you could say, "We are three of the tallest students in the room who are on the basketball team." "Room" is singular, but the word "who" here describes "we" and is therefore plural. For this reason, the verb should be "are."