Are birthdays holidays?
Or if not, is there a general term for all celebration days, including birthdays as well as public holidays (especially in AE).
Festivities? Festival days? Feast days? Revels?
A) Only if you're not going to work.
According to dictionary.com, which has the lengthiest and most inclusive list of definitions for the word that I could find, the following is the definition for holiday:
a day fixed by law or custom on which ordinary business is suspended in commemoration of some event or in honor of some person.
any day of exemption from work (distinguished from working day ).
a time or period of exemption from any requirement, duty, assessment, etc.
a religious feast day; holy day, especially any of several usually commemorative holy days observed in Judaism.
Sometimes, holidays. Chiefly British. a period of cessation from work or one of recreation; vacation.
Several other sources offer extremely similar definitions (see here and here) As you can see, the word holiday is virtually exclusive to days off from work, official or not. Having a birthday does not mean it's a holiday, and if you take the day off, you're on holiday or holidaying (and though this constitutes a holiday, it only applies to that one year, not ones in the past and future). If you don't take the day off, it's not a holiday.
It may be argued that any given day of the year has some kind of holiday or observance; this is correct but semantically I think you meant "is a birthday a holiday in itself"? The answer is no.
So holidays do not necessarily encompass birthdays. Meanwhile, a festival or celebration is only a birthday if the latter is recognized somehow, but according to this, that's the closest you'll get. Thus, there is no word that will incorporate both holidays and birthdays a hundred percent of the time, but that's just me being a prescriptivist. People will understand you if you call a birthday a holiday, festival, celebration, or occasion, so are they necessarily wrong? It's up to you to determine the usage in our fluid and multifaceted language.
Looking at Wordnet, the place where 'birthday' and 'public holiday' intersect is at 'day (a day assigned to a particular purpose or observance)'. So, as suspected, no, there is no general term for both that would serve your purpose.