I think those contractions are used more freely in European Portuguese. This advice from a Portuguese site: http://www.ciberduvidas.pt/pergunta.php?id=27735 says that they can be used even in formal situations, including academic texts, provided you use them consistently, that is provided you do not use "de um" in one place and "dum" in another.
As far as I know the moderator of the Duolingo lessons doesn't accept them, and I don't think you'll find any sentence using them here (except, perhaps, in the Idioms Bonus Skill).
"Em" is an uncontracted form (if placed right before a noun group, it means that noun doesn't take the definite article).
"No(s)" and "Na(s) are contractions of "em" with the definite article (o/os; a/as) - it needs to agree with the rest of the elements of the nouns, i.e., it is inflected according to the gender and number of the noun that comes after it.
Because the word you identify as "Monday" - we prefer to call it "segunda-feira" or "segunda" - actually takes the article after "em", all days of the week do.
P.S. There's always going to be "gender" in Portuguese words - every single one of our nouns is either masculine or feminine - what you may not have is the article that marks directly.
There are different opinions regarding the use of the article with proper nouns (which is what I think you meant by "Perfect noun"): I can't go into much detail, but for languages you shouldn't use it when you're talking about an attribute of the language ("something written in English") and not the language itself (English as "the English language").