Why is картошка singular
Картошка is uncountable, it always uses the singular form. It might help if you think of it as "some potato" rather than "a potato".
Any reason not to accept картофель?
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.
Why not сварите?
I'm pretty sure it's accepted. But if not, you can report it, because it's correct as well as "свари".
But the familiar 'Вера' suggests omitting the formal '-те'.
"Вера" is not a diminutive, its a full name, so it's not really too familiar. It's possible and quite common to use a name without a patronymic while still being on "вы" terms.
Does anyone else hear it like the Cinderella song? Золушка, Золушка, с утра до вечера, Золушка, "свари грибы", "и картошку", "мой посуду", "подметай"...
Вера, отвари грибы и картошку?