Ok, let me get this straight:
Обувь = shoes (any)
Кроссовки = sneakers
But then there's also спортивная обувь = sports shoes
Туфли : can also mean any shoes, but usually, one says "туфли бальные " = pumps (both for men and women)
And finally, there's ботинки: they are not necessarily boots, but seems to refer mostly to sturdy shoes.
Btw, here's a dictionary that also shows synonyms and many examples: https://tr-ex.me/translation/russian-english
In America, there is an English expression "Those are some big shoes to fill." It means that the person (who would be the one with the big shoes) did such a fantastic job of something, that they would be "a tough act to follow." Is "большие туфли" ever used that way by Russians?