What is the ruling on purifying oneself from urine after urinating, and if urine drops continue to be discharged thereafter, does that necessitate re-performing the prayer?

There is no need to expel residual drops of urine unless they come out on their own. Clearing the throat, walking, and pressing on the penis after urination to expel any remaining drops is an innovation not sanctioned by the Messenger of Allah, peace and blessings be upon him. The default is that urine stops naturally, and the more a person tries to expel it, the more may come out. Wiping with stones (istijmar) is sufficient, and washing the penis with water is not obligatory. It is recommended for one who has performed istinja’ to sprinkle water on his private parts to distinguish between its moisture and urine. If urine is truly continuously exiting, and not merely a result of waswasa (whispers/obsessive thoughts), and there is a time when it stops, then wait for it to stop, then perform istinja’, wudu’, and pray. As for one who suffers from urinary incontinence (sals al-bawl) that does not stop, he should perform wudu’ for each prayer and pray even if urine flows.