There are different type of worms. Dogs get tapeworms from ingesting fleas.
The de-worming medicine given to puppies...is not for tapeworms.
Here's everything you need to know about worms.
Identifying Dog Worms and Their Symptoms
Yes, multiple pets means everyone gets treated by flea preventative. To save a little on the cost...my three usually share a prescription vs buying one package each for all. Close guessimate on pricing...45 for a package containing 6 doses. If your other dog weighs more than 22 pounds...he will require more than one dose to treat. It's strength is given by weight. The cats would require flea preventative for cats as well.
I would bath puppy in dawn soap to kill any fleas he may have now. This method will not kill any eggs you don't get removed though.
If you have carpet....vacuum, vacuum, and I recommend sprinkling boriac acid all over and brushing in with a stiff broom. This treatment will kill in fleas in the carpet and will continue to kill any fleas which would hatch in 2 weeks. Wash all his bedding....
Good luck