στην vs. οτη are both correct. Why is that?
Strictly officially, only στη is correct because it is followed by β. Unofficially, there is nothing wrong to write it or say it in verbal speech, but that's my opinion.
I agree with Troll1995. Most people won't even notice if you say it, won't even notice if they say it themselves. xP
So βιβλιοθήκη is both bookcase and library?
Yep, exactly :)
I see you teached us the words "desk" & "chair" but I can't remember having seen the word "table"... why? Is there a special reason why you didn't put it in this section?
"table" will appear in the next Skill.