How deep into the game do you want to go? And what kind of rewards are you looking for? Just about every AMEX that has an annual fee has no FTF. Personal preference would be the AMEX Gold, but AMEX is less accepted in Europe so I'd also look at the Chase Sapphire Preferred or Reserve.
What Credit Karma tells you doesn't matter. You can get your actual score from AMEX. Under Account Services, Payment and Credit Options, View FICO Score.
Yes, if you downgrade before a full year, AMEX could claw back the bonus and theoretically even shut you down completely. If you downgrade at exactly a year, you will get a pro-rated refund of the difference between the two annual fees. I would wait until the next AF hits, then ask for a retention offer before downgrading.