I love the idea!
We have a doggie door in our screen door which our dog loves, under $300 from Lowes or Home Depot. I wouldn't put the hole in my main door either for the same reason.
My only thought on the door going directly into the house is whether or not the opening would be protected from the elements at all? When my main door is open and its just the screen door the super cold air really whips in and the heat in summer does too. So during severe weather months I open the main house door and close it as needed. But if you're in a moderate climate or like your temperatures it probably wouldn't matter at all.