Also, maybe I misread or received bad info, but if Burdi clears waivers etc., he has to be returned to us.
This is correct, but only for the next ~45 days. Rule 5 restrictions end at 1 full season or 90 active days, whichever comes last. Assuming he doesn't go on the DL between now and then, Burdi will have accumulated 90 active days by May 25th or so.
And in the unlikely event that the Pirates decided to cut him within the next 45 days, as you mention he'd have to clear waivers (with Rule 5 restrictions still attached) before he could be returned to the Twins. With less than 45 days left on his Rule 5 restrictions, I'm guessing he wouldn't clear waivers either.