The compound sentence joined by two conjunctions preceded by a comma is the last sentence: "The ability to access public spaces for sociable purposes is a critical component of the coming of age process, and yet many of the public spaces where adults gather—bars, clubs, and restaurants—are inaccessible to teens."
The word 'and' is not absolutely necessary. You could use the sentence with only 'yet', but the conjunction 'and' is mainly used to make the sentence flow better.