Is it necessary to look at the bottom of the feet when washing them during ablution to check for anything stuck to them?

This action of yours is correct, because the bottom of the feet is among the hidden areas that must be thoroughly traced, carefully washed, and cleared of any obstructing barrier, as mentioned by Zarruq. However, beware of excessive meticulousness in this matter, as it may lead to [obsessive-compulsive] whispering (waswasa).