How do the friendships between males and females relate to how Persuasion deals with gender overall?
When Anne goes to Lyme and the Musgroves meet with CPT Wentworth's friends, both males and females met and seem to have found a good sense of camaraderie. In fact, Anne claims that he could have made friends with all of them.
This seems to be an indication that Austen used youth and its carefree qualities to point out how the youths in the story do not hold the social friction nor establish the foolish boundaries of which Anne is so used to in her typical life.
It also indicates that youths also enjoy each other's best attributes in both genders. Notice how in their acquaintance the topics of marital obligation nor the pressures of society seemed to have played a role in their conversation.
Therefore, Austen uses friendship between genders as a way to filter the role of youth and freedom in how people communicate in the story.