I'm a Brit almost in my 40s and this is literally the first time I've heard of either of these words.
I'm a Brit too, and older than you are. I have heard the words all my life, and can assure you that they are not colloquial.
Well, obviously not everyone knows about it if a major newspaper has to write about what it is.
I can see how the sentense must sound odd but we say that all the time "Hvad skal du i pinsen/sommerferien/julen". If you want to throw in a verb i would suggest "lave". "Hvad skal du lave i pinsen".
They are synonyms, there is nothing grammatically wrong with saying "hvad skal du gøre i x", it's just that we never do.
Im an american catholic so i know what pentecost is but i have never heard of any kind of celebrations outside of mass centered around this religious event