Yes, "flea dirt" is flea excrement, and your cat had both fleas and tape worms. This is not unusual, this is not anything to panic about, most cats can tolerate both fleas and tape worms very well. Tape worms need fleas in order to complete their life cycle, so, since your vet killed all of the tape worms in your cat and the Advantage will probably kill all of the fleas in your house, you should be free of both problems. Before Advantage, living with the occasional flea and occasional flea-dirt was just a part of living with cats.
Getting rid of them is the best thing possible, and the treatment your vet did should do the trick.
However, I wasn't able to get rid of the fleas and tapeworms my cats had until I moved out of Florida. If you continue to apply Advantage to your cat at the rate recommended by your vet (I found out later that vets in Florida usually recommend Advantage treatments more often than monthy during the summer), you will probably be free of the problem and will most certainly significantly lessen the problem. Now that you know what flea dirt and tape worms look like, you will be able to recognize them in the future. If you eventually let your cat outside (even on a harness, I found out!) you can get a new population of fleas and tape worms.
But with inside cats in my new apartment, I haven't seen any fleas, flea dirt, or tape worms in an entire year. I have used no drugs since I carted my Advantage-treated cats and newly washed laundry out of the hot apartment.
It really depends on your apartment; my always-infested apartment had no air conditioning. The warmer your apartment, the harder it is to kill the fleas. If you've got a normal, temperature controlled apartment, only a few fleas, and only one indoor-only cat, Advantage treatment of the cat and laundering everything she sleeps on will probably kill all of the fleas.
I've sprayed my house as well, but that means that you've got to have a safe place for your cat to be while you poision the house.