To answer the question in your title, I'm not sure it is safe for a single woman to drive through Mexico. You may want to have a careful look at the official Government of Canada website....
https://travel.gc.ca/destinations/mexico
There are some areas that are NOT advisable to drive in. One other point to note is that if you travel in areas in which the Government has advised against travel, your medical insurance may be void.
My suggestion would be to board the Dog, fly down and limit your travels to normal tourist areas.

