Keep in mind that I have no personal experience with this. I'm just an information/research junkie who recently obtained Irish citizenship by descent, and have investigated what would be involved to bring my non-EU citizen wife to Ireland.
Your wife will need to
obtain a residence permit. I believe she can enter Ireland under a tourist visa, and then apply for long-term residence. Another person (charmender) commented on their experience
in this post. You might want to check with the Irish Embassy to confirm. It may be a good idea to have documents you will ultimately need for the residence permit, such as marriage certificates.
I was originally looking at this from the entry to Ireland standpoint. But after further investigation, it appears you should
formally abandon your U.S. permanent residence status.
You may want to keep in mind that U.S. citizens are required to pay U.S. taxes, regardless of where they reside. The U.S. has tax treaties with many countries under which double taxation is avoided or minimized. But unless your wife
renounces her U.S. citizenship, she will need to continue filing U.S. tax returns. The U.S. State Department charges a $2,350 fee to process these requests. Even if she does renounce, any income she receives from U.S. sources (Social Security, investments, pensions, etc.) will still be subject to U.S. taxes.
There is also potentially (depending on your recent income and assets) an
"exit tax" that may apply to both of you. I don't really know the details of that.
I'm not sure about the dog...