I'm sure this information differs depending on if you are in KDP Select (If you are, you can't publish it anywhere else. Period.) but from what I understand (from personal experience with having a perma-free short story on Amazon) that Amazon's bots scour the internet looking for any lower prices and what happens more often than not, is your book will just be lowered to match the price. If you outright tell Amazon it's available elsewhere (a trick used to get them to price a book free. This worked in the past, but could change anytime.) they'll usually change it to match. I'm not sure about Authors being told outright their books will be removed, only because I've never had that happen personally.
As for coupons, I don't have much experience with this, but I'd think it's probably OK. I have some short stories for $0.99 on Amazon and Smashwords which have coupons to lower them to free on Smashwords and the price has never changed on Amazon that I've noticed and I've never heard from Amazon about those stories being removed, BUT the coupons haven't been used in a transaction yet (they're only for Patrons to have free access), so I'm not sure if that plays a factor or not.
How Amazon handles adjusting/removing books seems to vary from situation to situation as well, so I can't say 100% what would happen if you were to offer a coupon on your site for a lower price. If you are really worried about how Amazon would react, I would say the safest way is to either go wide and not worry about Amazon, or simply keep your website price at the same as Amazon's without coupons.
Hope this helps some.