Amex has been generous with retention offers recently. You might want to try that first. I think you can even do it by chat now. There's a thread on Flyertalk if you want to get an idea what people are getting.
If you do end up canceling, the AF is refunded within 30 days after it posts. You can probably get another airline credit out of the Gold. Downgrading to Green, is that just to keep the account history? It doesn't have much benefits for the $150 AF. Also, if you've never had the Green before, you'd lock yourself out of the bonus.
As for pop-up avoidance, it's a mystery. It should help that you've kept the cards past the 1 year. If you're planning to apply for anything, would be better to do it before you cancel anything. My guess is it's more spend than card cancellations, but just a guess.