Is 14th of February vierzehnte zweite or vierzehnten zweiten?
With the usage of the name of the month, is it vierzehnten Februar or vierzehnte Februar?
Why is there an -n sometimes and sometimes it isn't?
It can be nominative:
So you see, all four cases can and do occur. And there are more prepositions besides am that can go with a date. Note the terminal -s in Februars for the genitive*, note the terminal -n in the adjective vierzehnten for dative, accusative and genitive.
Finally, another nominative form without an article is
*However: Not marking the genitive case of the month with a terminal 's' is becoming increasingly popular; at present, it is a matter of subjective choice.
The words are adjectives (literally, fourteenth second) in masculine gender, because Tag and Monat are masculine, and follow the usual rules for adjective declension. Therefore, it is:
In many cases, it is more common to use the name of the month instead:
Still, the same declension rules apply to the day.