Can't you say Mit dem Hund spielt seine Katze?
That would be grammatical but it would stress that it is the dog, the cat is playing with.
I would rather not. Just accustom yourself to the word order: subject - verb - ..... like in english.
Why not "der Hund?" does "dem" always go with "mit" when discussing something being with something else?
"mit" requires dative and the dative form of "der" is "dem".
In dative case, definite article of masculine and neutral are dem, while feminine is der and plural is den