Mexico recently passed a law saying that your dog must be restrained in the vehicle. This past Nov. while I was in Mexico I asked about this at the East Mexicali Border crossing. They said YES must be restrained. My 100lb dog was in backseat but I had leash attached to him. They said "OK" So each time I came to a check point or when crossing border I just made sure to have the dog leashed at that time. I was also asked to take dog out of vehicle at military check points, All OK.
I have always (past 15 yrs) carried vet certificates for my dogs and I have never been asked for them. But know that if I do not carry them..thats when they will want them :)
Jo