Dog running with pet toy being chased by another dog.
dog is running and being chased by another dog
Dog running indicates that a dog is running; being basically involves running by definition, but 'and' is necessary because the fact that the dog is running was already repeated and the first sentence was incomplete anyway.  Oh, wait, so is the second one.
A dog with a pet toy is being chased by another dog.
Running is a paraphrase of is running.