1. His job was eliminated (or cut).
2. He was watching through the open window as the women passed by. (past progressive verb tense needs something to happen)
3. His dog was barking loudly asking him for a walk. (this sounds odd because of the two "ings" together). It would sound better to say "The dog barked loudly asking him for a walk."
4. His dog was pulling him (on the leash) into the park. (fine). You could also say, "His dog was pulling him by the leash into the park." "By" emphasizes the importance of the leash to the action of pulling. "On the leash" is okay, but would make more sense next to "His dog" as in "His dog on the leash..." This emphasizes that leash was on the dog. It's more passive.