If I remember correctly, this is a first time I come across a word that is an en-word in Swedish, and et-word in Danish. Does this happen a lot?
Unfortunately, quite often. I get tripped up constantly.
Shouldn't "do you have a moment" be correct as well?
"do you have a moment?" would be translated as "har du en øjeblik?" and while the two expressions are interchangeable in english, they are separate words.
Would'nt "Har du EN minut" be more right or is that just in swedish?
No, in Dano-norwegian "minut" is only a neuter word, while it is only a common gender in Swedish.
In Nynorsk you have options to use it as a neuter or a male word. However it spelled ein minutt/eit minutt in Nynorsk.
25 minuter ?