If you visit the site, put stuff in your cart and get all the way to the checkout page, you must know by that point who sent them to your site. If the customer then goes away and googles for voucher codes, it shouldn't change the source. When (if?) they return, you should know they are the same person, from the same original source.
How did the affiliates manage to steal these customers? It seems to me like your referral tracking is massively broken.
Affiliate networks require you put their code on the "thank you" page shown after checkout. Conditionally removing that code would be seen as an attempt to scam the network and get out of paying commissions owed. What you're proposing doesn't work anyway -- you don't know that they didn't come through an affiliate link a week prior; the cookie isn't on your domain, but you owe a commission for that referral.
Correct. There's a "blind" tracking pixel at the checkout page that will usually catch the sale in a "last cookie wins" fashion. So on top of the advertisers, anyone actually facilitating the sale by advertising is ripped off by the coupon sites. It's a major pain for the industry - the only surefire way to protect yourself is to get rid of drop either affiliate marketing or coupon codes.
It's not so much stealing the customer as it is rescuing the order. It may not be the most ethical of techniques but they are claiming a bounty for converting abandoned carts.
Like someone else said earlier, the order process is broken by encouraging users to leave the website and potentially forget the order.
How did the affiliates manage to steal these customers? It seems to me like your referral tracking is massively broken.