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"saudade" can be used in the phrase "matar a saudade" (literally "kill the feeling of longing", by for example, meeting a loved one who you haven't seen in a long time). Saudade can similarly be used without the connotation of a long time having passed, e.g. "já estou com saudade com você" (I already miss you)

It doesn't (source: I'm Portuguese). It is just an often repeated cliché. What I would say is that Portuguese culture has a stronger preference for expressions of nostalgia than the average.

How does nostalgia differ from wistfulness?

Wistfulness is a general feeling of sadness and yearning towards something not present, whether or not you've experienced it before. "He was wistful about the life that could have been his if only he'd said 'yes' when he'd had the chance."

Nostalgia is a distinct feeling of want for something in your past, often from your childhood. "His nostalgia for the video games of his youth coloured his perception of how bad they really were."

Since you brought it up, and now that I've thought about it for a while, "wistful" seems like it might be a better fit for "saudade", but I'm sure that that word has all sorts of meanings that wistful doesn't have.