why mojej not moja? According to the translation it is a feminin singularis. Do I have to use genitivus? Why? Thanks.
I do not think there is a reason other than używać needs object in genitive. ("mojej kuchenki" is singular genitive, and feminine)
Well, if someone is using your cooker, then I guess this is the way to say it.
Oni używają mojej kuchenki. - They are using my kitchen. cooker translate as "kucharz" or female - "kucharka"
"kitchen" is "kuchnia". Theoretically "kuchenka" could be considered "little kitchen", but that's an overintepretation.
Kucharz and kucharka are "cook", not "cooker". A cooker is a synonym of a stove. And as Wiktionary (see here: https://en.wiktionary.org/wiki/cooker) does notice that it is chiefly British, I guess "stove" should be the default English answer, as Duolingo focuses on American English.
That really seems impossible. Isn't it actually that you were supposed to choose all the correct answers? This is a sentence with "They", so there are at least two starred answers...