Re: Wife realizes now that she does not like living with my dog
They don't even have a child at this point, so I find the argument to be more than a bit premature. Sounds like a nice little smokescreen to hide the fact that she simply does not like the dog. Conceiving or adopting a child can take years. I see no valid reason to get rid of the dog on the grounds of "protect the child" until there actually is an ETA on said child.
Compromise is absolutely critical for a good relationship. Unfortunately, some things are simply not suitable for a compromise. One or both partners needs their way entirely. People typically cannot compromise on things like having children and relocating. You either agree to do them or agree not to. This dog situation is another good example. He has already compromised by retraining the dog, crating it, banishing it from certain parts of the house, etc. This has not softened her, she still starts fights over the dog. She will simply not be satisfied as long as that dog is part of their lives.
This doesn't mean that she is "wrong" as needs are not right or wrong (they are frequently incompatible, though.) She may be simply unable to compromise. In such a case, the other party must be willing to totally give in. Sometimes that's possible. Only you can decide if you want to accommodate her.
I will echo what I wrote earlier that I suspect there are a lot more fundamental mis-matches beneath the surface. You really need to explore your respective needs and desires before having/adopting a child. The fact that she so continually makes an argument about this suggests that she argues by war a war of attrition. She will keep making you miserable about this until she gets what she wants. That, my friend is not a pleasant way to live.
I wish you luck.