Could one write: Moje urodziny są w tym miesiącu (?)
as you probably know by now your sentence is My birthday is this month. while duo's My birthday is this month. but both are perfectly normal and common.
urodziny is plural?
Yes, grammatically plural, plurale tantum.
sa is the plural form of "to be" as in English "are"...how does it work in this sentence when the verb is singular?
Birthday is singular in English, but plural in Polish. It has no singular version, it's plurale tantum.
"mam urodziny w tym miesiącu" should be accepted
Well, it changes the subject completely (from "My birthday" to "I") and there's really no need to do that as both sentences translate very easily and directly.
I don't like the idea of changing the word order in the translation. There is a change of emphasis in English as well as in Polish.