I know "-ino(a)" is the diminutive ending, so perhaps copertina? I don't know, though. Gotta get a native on this.
See, now I'd recently learned that everything is funnier in Duolingo when you type your answers in Caps-Lock. This is certainly no exception.
Do you have to say "la mia coperta" or could you just say "mia coperta?"
I think you have to keep the article because it's an object, you're usually only able to take it away when referring to family members. Just a guess though.
You're correct, if I remember correctly from my college Italian classes. Also, if you're speaking to the noun modified by a "mio/a/e/ei" then you can leave it out: "Grazie, mio amico!" And you have to leave it in with family members if you make certain modifications, but I forget what they are. If you reply to this comment to ask, I'll look them up from my textbook for you, but I think they're things like plurals require the article (mia sorella but le mie sorelle), slang requires it (mio padre but il mio babbo), and suffixes require it (mia sorella but la mia sorellina). I could be wrong about some of those though.
I believe you're absolutely correct and the examples you cite for use of the article are the ones I'm familiar with.
So, I practise historical fencing, and our club mainly does the Italian tradition of fencing. One of our high guards (used to protect yourself from cuts coming from above) is called a 'coperta', in the sense of a 'cover' ('coperto' actually means 'cover' in modern Italian), or 'covering guard'.
You can imagine my puzzlement when I read this sentence, haha.