"She is buying nineteen blue plates."
Translation:Ona kupuje dziewiętnaście niebieskich talerzy.
Yes indeed. Most numerals actually take Genitive..
The ones that take simple Nominative are: 1, 2, 3, 4 and those ending with -2, -3, -4... apart from those ending with -12, -13, -14. And that's just the beginning of Polish numerals, because then of course the function they serve in the sentence changes the case of the whole 'numeral+noun' phrase...
You might need to say this sentence aloud, instead of writing it (although that is pretty unlikely I admit).
In the French course you can put the numerals instead of spelling out the words and it is accepted. I always do this for numbers and times when I am writing the English translation of the French text since I don't need the practice at learning "a quarter to five" is the same as "4:45". You could check and see if this works for the Polish course too.
As far as whether a Pole would bother to spell out the number we'll have to wait for a native to say. It's probably the same as in English though, sometimes you would and sometimes you wouldn't. It depends on the writing style and personality of the writer. I spell out numbers as often as I use the numerals when writing something that is primarily word based with a number here and there.
that depends, I prefer to write the whole number/amount in more formal things. I think it looks better. Also you have to write "Słownie" on any money related documents.
And you know only situation where I can imagine anybody writing down "Ona kupuje dziewiętnaście niebieskich talerzy." are : texting, and some kind of story/novel. SMS 100% use digits, books 99% use words