No
Dwi isie mynd mewn means "let us enter".
If the warrant shows up in the national crime database, and they run your name, I believe it is quite likely you will be.
You only need a valid passport to enter Mexico.
Mexico
Having a passport and/or Visa is a legal way to enter the U.S. from Mexico.
Your birth certificate will not help you. You can enter the US from Mexico with an "enhanced" US drivers license, but only if you are traveling by sea or land, not by air.
A US citizen with a DWI conviction may be deemed inadmissible to enter Canada. However, it is possible to apply for a Temporary Resident Permit (TRP) or Criminal Rehabilitation to be granted entry to Canada. The application process and requirements can be complex, so it is advisable to consult with an immigration lawyer or seek guidance from the Canadian consulate or embassy.
As long as you have a US visa with you, you can also enter Mexico.
Yes. They need a warrant unless you give them permission to enter. It does not matter if you know the person or not, they either need a warrant or your permission to enter your house to search for anything including a person.
To gain Arizona and New Mexico from Mexico
Yes, assuming the person will follow the required legal procedure. An individual needs a passport to enter Canada from the US. Because of National security issues all persons crossing into Canada, Mexico or leaving US borders are subject to extensive scrutiny. The outstanding warrant would appear as soon as border guards checked the individual's ID.
probably notAdded: Highly unlikely - especially in this day-and-age, all lawful incoming passengers to the US are screened through a database, and if your warrant appears there, you will either be detained and held for law enforcement or denied entry to begin with.