"They wrote each other every week."
Translation:Psali si každý týden.
It is "each other" and "every week", the "other" belongs to "each" not to "every". What you suggest would be "They wrote each other every other week."
Also, "každej druhej" is Common Czech, are you sure it is supposed to be accepted? If yes, then "tejden" is in order to stay consistent. But in written form it looks like 18th century, it is normally only spoken.