Am I allow yo be inside the store with my regular dog (not service dog, he’s not that smart lol) ?

Unless it's a Service/ Medical Guide Dog HEB will allow a dog or any other pets, the other situation some HEB may have Veterinarian Office on the lobby only then pets are allowed for that section of the Lobby

Yes, I doubt they will say anything to you as long as you are carrying him, or he's on a leash. Just remembered if anything happens, your responsrable.

Thankfully no, absolutely not.

I hv seen small carry on ..in purse or in arms dogs in the store... and working service dogs but not a regular size dog on a leash or otherwise.

Hi Maria - not to my best knowledge.