"Vera, boil the mushrooms and the potatoes."
Translation:Вера, свари грибы и картошку.
Картошка is uncountable, it always uses the singular form. It might help if you think of it as "some potato" rather than "a potato".
No reason, except it is a formal word people do not actually use in speech (it is something you would read on a price tag). We prefer to stick to картошка, which is what you can hear.