"There are a dog and two cats in his house."
Saying "are" then "a" just sounds wrong. I would not expect to see this when reading an article or watching the news. However, a writer of a story might want to give an image of a dog and two cats together as a group. When we look at it this way, an argument can be made to use plural.
For example, when we refer to sports teams we use "are" instead of "is":
Q: Who IS winning?
A1: The New York Yankees ARE winning. / NOT The New York Yankees IS winning.
(Even without the "s" to make it plural):
A2: The Boston Red Sox ARE winning. / NOT The Boston Red Sox IS winning.
If you are going to use "are" then give a pause like a comma before saying "a".
SHORTENING THE LANGUAGE
Another point I would like to make is that native speakers shorten their language. I think I would hear "There's a dog...." more often than "There is a dog". "There's" is a common default, even if we switched the order. "There's two cats and a dog in his house."