Translation:My entire family needs a refrigerator.
Weird sentence. The word entire/ whole implies an extended family of some sort. Yet only a single refrigerator is needed. For single household families I don't think you would stress the 'whole' family. Just 'My family needs a fridge. '
"czarny mój pies" would sound like from a poem (probably one for children, given the topic).
As for "cała moja rodzina"... I feel that I'd put it like that because this is "my family" and also it's "whole", rather than "whole family" which is also "my". That's my feel, I can understand if someone won't agree with that and we do accept both versions.
I guess it's not only mine feel though given the fact that it wasn't me who wrote this sentence ;)