Translation:Please put this paper on the fridge.
1) Is れいぞうこ the common word for fridge?
2) Reading the translation of (はって) はる being "to place", I first tried to make a sentence with the meaning of "put the paper in the fridge" (wouldn't have been duolingo's weirdest sentence...) before realising the actual meaning is more like "stick the paper on the fridge". So, is it even possible to use はる to make a sentence like my first thought was?
1) Yes, as far as I'm aware, れいぞうこ is the only word for fridge.
2) No, 貼る (はる) means "to stick on" or "to affix", rather than "to place", though "to place" can be used as a translation in some situations but not all. For example, you could "place" the paper on the front of the fridge which means "to stick on", but if you "place" it on top of the fridge, you wouldn't use 貼る. Similarly, "to place (something) in" uses a different verb, 入れる (いれる) "to insert".
Alternatively, if you specify れいぞうこの中 (なか), your sentence becomes "stick the paper on the inside of the fridge please." Although, while the paper ends up inside the fridge, in Japanese, as in English, the sentences are different.
Bug? When I had to assemble this sentence earlier, the English version of the sentence was "Please PASTE this paper on the fridge", but when I had to translate from Japanese now and used the previous translation Duo provided me with, I got it wrong. It would be great to see this fixed just for the sake of consistency :)