Does an extremely small amount of liquid discharged shortly after urinating invalidate ablution, and is it necessary to check for its discharge after every urination, and what is the ruling on sprinkling clothes with water?

If the discharge of the aforementioned fluid is real and not imaginary, then it nullifies ablution, for ablution is invalidated by anything that exits from the two private parts, whether little or much, pure or impure. There is no need to wait after performing istinjāʾ. As soon as the discharge stops, you perform ablution and pray, and your prayers will be valid. The default is that no discharge has occurred, and the ablution is valid. It is recommended to sprinkle water on the private parts and trousers after ablution to ward off waswasah (whispers of the devil/obsessive thoughts).