Your first question's answer: YOU MUST USE A FILTER! How do you think you can make good, quality coffee on any coffee-maker without a filter? It lets the good stuff go through and keeps the bad stuff out (if it is high-quality or just decent enough).
Your second question. So siphons so use metal filters, but they are either rare or even not produced anymore. Your cloth description is right:
A piece of cloth can however get dirty, rough, etc.
if you don't know when or how to clean it. In good quality plus cleaning, they can be used up to 100 times or even more. In this site (which states all types of filters and how to change them), it says that:
My cleaning regimen for cloth filters has changed over the years. Cloth filter assemblies should be cleaned right after use with a soft brush and running water, and then soaked in a solution of oxyclean and boiling water. Rinse well after, and store dry.
This will not harm the cloth like bleach will, and it comes out sparkling white and odor free.
Oxyclean is a product that is like washing detergent. Cleans up cloth and fabric quickly and nicely, bringing out good results. Boiling water will help out with the cleaning and disinfecting of the cloth. It will literally burn and kill any harmful bacteria on your cloth. You should clean the cloth immediately after use. This will keep the cloth in top-shape for a long time in its "life" span.
Paper filters' cleaning ways are obvious. One use and they are thrown right into the trash. You really can't clean paper filters.
Cleaning glass filters (in my opinion) should be cleaned right after used. They are pretty rare and easily broken when not receiving enough care. Just clean it like any other glass product. Put water on a paper towel and wipe it clean.
The rough (but not sharp) surface of the middle was the filter - coffee grounds would get trapped in the little channels between the bumps, but liquid would still pass through. In practice, these glass filters allow a fair amount of sediment to pass into the brewed coffee. Some find this beneficial, others do not.
The choice of using this filter is your decision depending on what you want from your filter.
Nylon mesh are easily cleaned: they're dishwasher safe. Yet, they are breakable and fragile at the same time. Be careful with these guys.
I hope this helps you and answers your questions!