Are small pebbles that enter one's shoe considered impure, given that roads in India may be contaminated with urine and other impurities? And is it obligatory to wash the shoe, foot, and trousers upon returning home?

There is no need to wash shoes, feet, or the bottom of trousers due to doubt about the purity of small pebbles, because the default state of things is purity. Even if they were impure and dry, they would not defile a dry place they come into contact with. Therefore, this washing is not legislated.