These exercises are all rather reliant on my (not fantastic) ability to spell the days of the week in English...
No; nicht cannot be used with the indefinite article. It is just one of those things you have to learn and accept ;)
if you had to submit a report by Thursday but it took you a whole month to do it and your boss yelled "Thursday is not a month!"
I understand why Donnerstag ist kein Monat translates to Thursday is not a month. But suppose I want to use the stressed translation Thursday is no Month. Does that back translate to kein Monat or does German deal with it in a different way?