Life is not about domination, it's about compromise, especially in marriage. Take this advice from someone who has been married for 39 years.
Your wife was wrong to get the puppy, but she probably thought you'd change your mind when you saw how happy the kids were. Her actions were manipulative, but there is a more important issue here. She was probably thinking about the kids and how much she loved the puppy. She rationalized that you'd come around.
Your reaction, on the other hand, was petty and vindictive. Pound dogs are put down. Kids at school will tell your kids this and your kids will NEVER look at you the same way again. Your wife probably won't love you the same way again... ever. You have made a really, really big mistake. But it's not too late to fix part of it. Go back to the pound and bail out the puppy. You might have to pay the adoption fee, but that's minor at this point. Bring home the puppy and an adult-to-adult conversation with your wife. If she really, really wants the second dog, let her have it.
Your concern at this point should be about your future relationship with your children and your wife. At the very least, you can find the dog a foster home until it can be adopted. Look into pet rescue organizations. My wife and I have fostered 7 dogs and cats. One of our foster cats went to a little kid who was autistic and the cat really changed his life.
Sorry to be so direct, but there are so many juvenile responses here by kids who are trolling or just don't understand life yet. Your wife made a mistake. You have made a bigger mistake, but it's not to late to make it right. Best of luck with this.