Interesting. Didn't know that czy could be used to mean "or". I'm familiar with lub and albo.
"czy" is often used in questions,
When would you use "albo"?
in a statement. When only one is possible..
Could "bin" should be included as kosz as is, or is it necessary to state in full "kosz do śmieci" to mean bin or dustbin?
„Kosz” is just general term, could mean anything from basketball to bin, but if context makes it clear Poles are frequently using only „kosz” to mean „kosz na śmieci”, yes.
Wait, I speak polish myself,but, "Kosz" means trash or trash can. But it can be used as basket.
I entered "kos´" (i.e. without "z", otherwise correct) and did not get a hint that this is not correct. This could (and should) be improved.