You can say "don't you know", and "do you not know", but you cannot say "do not you know" even though that's what "don't" means. I don't have an explanation for this.
Yeah, now you mention it, that's a weird one. I've never even noticed that. :)
Can you also say "you don't know your own password?" or can "swojego" only be translated as "your" (not "your own?" )
Technically, "your own" is "własnego" (or "swojego własnego" to give even more emphasis). Although if you emphasize "swojego" with your voice, it can be enough to denote 'your own'...
I find it difficult to understand the use of "swój, swoja, swoje" "Would it be also correct to write "twojego"? Is the Polish for "I forget my password "zaponiałam swojego hasła"?
"twojego" would be technically correct, but kinda clumsy. If a form of 'swój' is correct in a given sentence, you should use it rather than the 'normal' possessive.
Yes, "I forgot my password" would be "Zapomniałem/Zapomniałam swojego hasła". "swój" always refers to the subject of the sentence.