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These are likely roundworms. These are the worms that hang out in the stomach and are most likely to be vomited. They look like spaghetti noodles, and they'll curl up when they come out. To treat them, we use Panacur (Fenbendazole) for 3-5 days.
There may also be lung worms or even roundworms that have migrated to the lungs. That could explain the coughing. Panacur will take care of those as well. It could also be kennel cough or some other infectious respiratory disease. Those are treated with an antibiotic- Doxycycline.
I hope this helps, let me know if you have any other questions.