Thanks for the quick reply.
1. While I cannot diagnose this over the internet it sound to me like you have been seeing tapeworm segments, you can usually see them round the cat's rectal area or in their faeces, they are about the size of a grain of rice and you can sometimes see them slowly move.
Here is a link to a picture of tapeworm segments : IMAGE
2. This is no cause for concern, this can be treated with tablets which your vet can supply or you can get them over the counter from a good pet store. Try something like Tradewinds which is now an over the counter product and is the same as Droncit which before you could only get from the vet, it contains Praziquantel. I have prescribed loads of Droncit over the last twenty eight years and found it to be safe and effective. Tape worm treatment will be different from regular round worm treatment which you can also pick up at your vet or pet store.
3. Make sure your cat is free from fleas as they can be part of the tape worm life cycle.
4. Cat tapeworms are very rarely a risk to people, here is a link which covers the subject : LINK
5. It would be good practice to worm your cat regularly against both tape and round worms.
If I have not answered your question fully enough or if you would like to ask more I will be on line for the next hour or so and I will be glad to discuss your question further.
Scott Nimmo BVMS MRCVS