Whoops I forgot to say that your vet should be able to either give at the clinic or send home the prescription of Drontal (or whatever broad-spectrum wormer they use) for you to give. And dogs, Heartgard or Iverheart (worming for heartworms) usually worm for other types of worms every month. Luck!
Very very unlikely. It would be fecal transmitted. You can get roundworms from a cat, which in rare cases results in eye lesions, but tapeworms....eh. You probably don't have to worry. If you have young children worm your cats and dogs often.