You asked several questions. Let's start with the original sentence:
I want you to put every new word you learn down onto sticky notes, and stick them all over your house so to better memorize them.
The answer to the title question is yes: to stick is fine here. The meaning is close to that of "to place." You can also stick things up: this would add a sense of the object being placed somewhere in view and of it being held in place by something against gravity. In this case it's fine with or without "up."
By the way, was the phrasal verb put down used correctly?
It's not wrong, but I think we could make some improvements. First, I would keep the preposition next to the verb: "... put down every new word you learn onto sticky notes." Next, I think the sense added by "down" is redundant with the sense added by "onto." You don't need two prepositions to do the same job. You can just write "put ... onto sticky notes." Answering your last question, I do think on sounds more natural than "onto." Finally, there is a semantic conflict between the grammatical number of the words (every new word is singular) and the number of the sticky notes (plural). This is a semantic but not a grammatical conflict.
- I want you to put every new word you learn down onto sticky notes (original)
- I want you to put down every new word you learn onto sticky notes (better, moved preposition)
- I want you to put every new word you learn onto sticky notes (better still, eliminate redundant preposition)
- I want you to put every new word you learn on sticky notes (better still, simpler preposition)
- I want you to put every new word you learn on a sticky note (best)
- I want you to put all the new words you learn on sticky notes (also fine, if you prefer plural)
Finally, although you didn't ask about this, I think "so to better memorize them" is ungrammatical. It should either be "so as to better memorize them" or "to better memorize them." And "more easily memorize" has a more correct meaning than "better memorize". But the whole clause sounds just a little awkward, so how about:
I want you to put all the new words you learn on sticky notes, and stick them all over your house so you can memorize them.
I think most of this is pretty advanced-level nitpicking.
