This sentence is much too wordy.
How about --
Heidi liked to play with her dog, Rocky.
Is the sentence correct?
Heidi had a dog, and he was named Rocky, and Heidi liked to play with him.
Answer: I think it should be
Rocky was the dog's name, he belonged to Heidi, who liked to play with him.
2 answers
In addition to being terribly wordy, your correction includes a run-on.
Ms. Sue's sentence avoids the run-on AND the wordiness!
Ms. Sue's sentence avoids the run-on AND the wordiness!