"перо́" is a feather and also the colloquial name for a fountain pen (перьева́я ручка) and a stylus for a digital graphic tablet.
It depends on the situation. If you are using something as an instrument, you don't put с as per the answer.
If the subject is actually -with- something, you use с. For example, с гривами is "with mushrooms."
Does the use of the instrumental case make it "a pen" (in general) instead of "the pen" because "the pen" was marked incorrect.
Yes, but ручка is a diminutive form of рука. Still, formal language should be used, at least with Duo on Duolingo, and it is very unlikely that you would draw only with your arm/hand.
Unless you are a very small child, you won't say "я рисую ручкой" when you mean your hand, and not a pen.