Where in the Seller Central product listing section marked "KEYWORDS" should I place keywords to attract the best search results for my product beyond its what people might call it by name or its description? (Click this post for a more detailed explanation of what I mean)

Use the "Search Terms" field. You get 250 bytes of space which generally means 250 characters.

Amazon limits the length of the Search Terms attribute to less than 250 bytes. The Search Terms limit applies to newly-registered as well as existing ASINs.

https://sellercentral.amazon.com/gp/help/23501?language=en_US&ref=ag_23501_cont_10471

Lots of good info on that page about the dos and don'ts.

Adding new keywords to the listing shouldn't affect the organic results for the keywords that the item is already ranking for.

You can do PPC ads and use the same keywords within the same campaign. I don't believe there is a reason to have two different campaigns for different target uses with the product. You might still want to do two PPC campaigns though. An auto campaign that you constantly monitor. Take the good converting keywords from that campaign and through them into a manual campaign. Then add those keywords as a negative keyword to the auto campaign so that the budget for that campaign is used for finding new keywords and appearing on other ASIN pages.

/r/FulfillmentByAmazon Thread