Hi,
Is "3 days and 2 nights trip" a right way to speak and write in English? No.
Or should I write it as "three-day-two-night trip"? If you want to say it that way, I'd write it as
'a three-day, two-night trip'. But I'd prefer to reword it, perhaps as 'a trip for three days and two nights'.
Best wishes, Clive