Well done, Yuu! A very good attempt, and through help of internet. It is definitely understandable, but as you suspected it is not quite correct norwegian, even if it gets the meaning through very well.

Please ignore the following attempt at typo and grammar correction if it is not wanted (it will by stuffed with english typos and grammar mistakes, I'm afraid):

Jeg ryddet skrivebordet i går. Jeg la papiret i papirsøpla og organiserte innholdet i skuffene. Nå er notisbøkene og tegneutstyret/kunstmateriellet stilt opp loddrett. Skrivebordet mitt er veldig rent og ryddig. Jeg er fornøyd med det.

Norwegian prepositions will kill you because they Are Not Logical (or the logic is hard to spot). In this case, "rydde" is better than "rydde opp". "Rydde opp" is something you do to rooms, or to clutter: Jeg ryddet opp på kjøkkenet" or "Du må rydde opp i dette rotet".

"legge" is preferable to "putte" since "putte" indicates placing something inside something else, like stuffing things into your bag). You could also very well use "kaste", if the act of placing the paper in the recycling bin is not as important as the fact that you got rid of it. So an alternate way for that sentence could be "jeg kastet papiret til gjenvinning"

Any Norwegian would understand "gjenbrukekassen" but it isn't actually a word in use. I mostly use "papirsøpla", but that is sort of . "Kildesortering" is the of sorting of the garbage. "Gjenvinning" is recycling.

"Stakket" is not a Norwegian word, I think you might be looking for stablet, buuut that word in itself means to stack horizontally, so I think stilt up feels more idiomatic.

Renslig=cleanly. More commonly used for when the puppy has learned to do it's business outside instead of on the carpet. Rent=clean.

You could use "tilfreds" as well, but "tilfredsstilt" means more that someone else have made you so, like gratified, I think.

And woo, organized desk!
Una