Yes, this is possible because modern medicine is so advanced now that everything is possible. We should have the cure for cancer and AIDS, and in due time, we will. Researchers are experts in their field and can find a cure for everything. All it takes is a little funding.
Some of the less strong painkillers have very few side effects or only have side effects in large doses. It is possible, as scientists are working hard to eliminate the side effects from drugs. It would be a great thing if they could find a pain killer that had no side effects.
Painkillers without addictive properties or do not cause the patient to become drowsy or disorientated are already on the market. However, these painkillers tend to be simply adequate at providing pain relief and do not tend to help patients suffering from extreme pain very much at all. While it is perfectly possible that a painkiller without side-effects could be synthesised, only time will tell whether the medical establishment will consider it worth the trouble.
If the cells can't use it to build then it will be filtered through the liver as a toxin. There are millions of chemical compounds out there and only 90 or so you're body can actually use, rest your body filters out as a toxin via liver. The only way a drug could possibly have no side effects is if you decide to call all the side effects "main effects". The human body is way too complicated for any chemical to only have one effect, hell, even sugar has "side effects" (hyperactivity, weight gain, cavities, increases risk for diabetes, etc.)