Is it a standard practice to sever sympathetic nerves to treat unexplainable pain?

Last resort. The phenomenon you mention is crps aka rsd, a type of sympathetically mediated pain. Cochrane database did an analysis in 2010. Basically concluded that data is limited. However, may be tried as last resort if meds, and other techniques and devices failed and still major disability. See pain specialist for guidance.
Yes. The procedure is known as sympathectomy which is cutting the nerves to treat unexplainable pain in the arms or legs. The procedure is not performed very often.