Having the dog is one single detail of the work environment and would neither make nor break it. I fail to see how this is some sort of huge revelation that shows that she's a liar.

However, I do think it's rather unprofessional if she's spending your sessions whining about her work environment. Not only should your sessions be about you, but it's not really appropriate for her to be sharing all the complaints she has about the facility and management with you. It's just unprofessional.