Old Dogs With New Game
You only get to take care of your dog for three weeks, then you cannot play the normal game. You must save the file to "Always Together Mode". When this is done, you can start a new game and still keep your old dog and file.
Get All The Dogs
Adopt a dog, then retire it as soon as it is in the playpen. Since its happiness will be in the green, it will retire a happy dog. Repeat this to get all the dogs in about ten minutes.