Happy BirthdayS, Duolingo: you started your private beta on November 30, 2011. On June 19, 2012, you were launched for the general public Then, on November 13, 2012, you released your ios app through the itunes app store. On May 29, 2913, you released your Android app. Duolingo has about 200 MILLION registered users across the world...¡Feliz cumpleaños, Duolingo!
Noun "cumpleaños" is both singular and plural form, because the Spanish nouns end in -s that are not stressed on their last syllable don't change in plural. For example: lunes, martes, paraguas... If a noun ends in -s, but it's stressed on its last syllable, it gets an -es in plural. For example: autobús > autobuses...
05/26/18. Yes, yesterday was 05/25/18. Did you mean to leave a reply that was responsive to my question? Have since realized that there is no "cumpleaño" in Spanish. But rather "cumpleaños" is both the singular and plural form, with only the article changing to indicate whether "birthday" or "birthdays" is intended.
12/18/18. Excellent first question. Have searched and am unable to locate an authoritative source that answers this question specifically for "feliz," but I may here be able to back into an answer that will both (1) be acceptable until others are able to cite to an authoritative source directly on point and (2) provide an answer to your second question.
Looking to the examples provided at "http://www.spanishdict.com/translate/feliz", it appears "feliz" is usually placed after the noun it modifies (like the overwhelming number of Spanish adjectives), but see also one example given: "[T]uvo la feliz idea de reservar en otra aerolínea." (He had the happy idea to book on another airline.) Thus, based on the examples, we know at least that "feliz" is one of the relatively few Spanish adjectives that can sometimes be placed either before or after the noun modified.
Beyond the above, if you are looking for a 'grammar rule,' I would also hazard to say "feliz" is placed before "cumpleaños" here to shift the emphasis of the phrase to "happy." Butt, John and Carmen Benjamin, A New Reference Grammar of Modern Spanish, § 4.11.3 (Adjectives put before a noun to indicate impression, reaction or subjective assessment), p. 62 (4th ed. 2004)["The most common reason for putting an adjective before the noun is to emphasize its emotional content."]
Finally, one may simply remember the adjective placement because "feliz cumpleaños" is a commonly used 'fixed expression.'