"I don't have this kind of problems." This can't be proper English grammar. It sounds so wrong.
It's proper, grammatical usage. Google "I don't have those kinds" (with quotation marks) to see plenty of valid use cases.
I agree, but the solution Duolingo gave me was "I don't have THIS kind of problems."
It'd have to be "I don't have these kinds of problems." What Duo gave you is indeed incorrect.
I understand that the Russian is literally plural, but to me "I don't have this problem" is much more idiomatic in English. Am I right in thinking that "У меня нет такой проблемы" would be less idiomatic in Russian?
I don't know about Europe, but in English, we are accustomed to saying, "I do not have those/these problems."
You have a referent, and such does not refer back as well as those/these.
Further, these and those are plural ... such is used more often in the singular.
is this the ONLY possible way to say this ? I mean, only with the genitive plural ? or can we use the accusative plural ?
When 'нет' is used to describe something that doesn't exist (in this case problems) it takes genitive. "У меня нет кошки." Here 'кошка' takes genitive because "There is not" a cat.
Both the Russian and English translation are plural. I think that is how this question was designed.