I did not have a dog/pet whatever following me around when I entered my Markarth house one day and my child asked to keep the nonexistent dog. I said yes (confused) and nothing happened. No pet was there. I want to be able to actually have a pet for my child. I attempted bringing Meeko to her but nothing happened. She didn't acknowledge the dog in any way. I also tried to buy a dog from the war dog seller outside of Markarth but he said I already had a pet (before Meeko started following me). I can't remember if I previously had a dog following me around and it died or not (I could be thinking of my past save file on a different hard drive).

Anyways 1) what is happening? 2) can I adopt another child and show them a dog as a pet (even if my first child already has a nonexistent dog)?

1) This is the bug with the stray dog. More info here: http://www.uesp.net/wiki/Skyrim:Stray_Dog#Bugs

2) Although there is no way to fix the vanishing stray dog bug, you should be able to adopt another child and "show" them another dog (such as Meeko).

Thank you for your swift reply!