Shouldn't "We don't travel in the winter." also be accepted?
It should, but there's not much that can be done now besides report it. Hopefully at some point these reports will be addressed.
Is '-ку' in 'взимку' a case marker or a suffix '-к-' + the case marker '-у'?
Already reported it, but "...THE winter," should be acceptable here. One of the most common mistakes Ukrainian and Russian speakers make in English is to omit articles, "a," "an," and "the," which don't exist in Ukrainian (or Russian).