Immigration proof of payment?

But since they had a stamp, and were flying out, how do the people at the Alaska airline ticket counter know whether or not the visa has been paid for or not? Do they always ask for proof of payment when you don't fly in with them? I've never been asked before....???

Probably if the "entering by" section is checked "air", they assume it was paid by the airline. If it is checked "land", they may request proof of payment.

It's been almost 10 years since I entered Mexico by land from the US, so I don't know what the current procedure is there, but I know for certain that you can enter Mexico from Belize and get your FMM stamped, and drive into the country, without ever stopping at Banjercito. At the crossings here, they check for proof of payment when leaving for Belize. That's why I can say that it is not universally true that a stamped FMM means you have paid the fee, because it is not the case at all crossings. I bet that is just the standard procedure at the busy northern crossings, but not an official regulation. That's just speculation, though.

Leaving for Belize, they'll hit you up for proof of payment of the fee, even if you arrived by air. You need to bring a copy of your itinerary showing the taxes & fees paid, if you wish to avoid "paying twice".

I guess the key take-away from this is that if you're doing anything unusual, such as flying in and leaving by land, or vice versa, then you should keep your receipt/proof-of-payment and be prepared to show it when leaving. But I still don't think it was an illegitimate request or scam.