Does cheap lens filters (UV, ND, CPL) affect on image quality, color reproduction, sharpness and etc.?

All filters affect image quality in some way or other. Some effects are desirable, and these are usually the reason for using a filter in the first place, and some are undesirable. Inexpensive filters are usually built to a lower quality standard than more expensive ones, and so they often have more undesirable effects. Low-quality filters are more likely to cause problems like lens flare, distortion, unwanted color cast, and vignetting. Better filters are more expensive because the features that help avoid these problems add to the cost of production. For example, multi-coating both sides of a filter adds a number of steps and requires extra equipment and materials.

And do cheap UV filters really do protection job?

Yes. Physically protecting the front element of a lens is the other reason that people typically add a filter to their lens. It certainly does work: with a piece of glass in front of the lens, the front element is protected from dirt, dust, moisture, fingerprints, etc.

There's a bit of a tradeoff here, though: you don't want to spend a lot of money on something that's intended to sacrifice itself to save the lens, but you also don't want to put something in front of the lens that's going to adversely affect the photos you're taking. There's not one right answer; in choosing whether to use a protective filter, you need to consider such things as the cost of your lens, the cost to repair any damage to your lens, your budget, the environment in which you use your camera, and the down side to whichever filter you're considering.