When I had a dog, (last trip for my dog was 2013) I was stopped about 50% of the time crossing at Sonoyta, I had a large dog who would always sit up at the border, even if I had a green light
they would pull me over when ever they saw the dog's head pop up. Never happened at night, but I seldom travel at night now.
They want to see that the rabies vaccinations are up to date. I only presented a receipt from my vet showing the expire date.
Last 2 trips with my dog, I copied the vet receipt along with a photo of my dog and my drivers license all on it.
It sped up the process at the border and the Mexico agents were very happy I did this because it saved them some time and work.