Why doesn't "espejo" always need an article?
I know that the placement of "un" or "una" (or "el" or "la" for that matter) in front of a noun depends on context, and that often articles aren't necessary.
However, "espejo" seems to be a special case; for instance, "No tengo espejo" is an acceptable translation of "I don't have a mirror" even though there is no "un" in front of it, while the correct translation for a sentence like "I don't have an apple" seems to require the use of "una" before "manzana".
Is this how it is for everything, and Duolingo just happens to be more lenient with "espejo" than other nouns? Or is there a specific rule that dictates whether an article is required?
It occurred to me that there may be different, "unspoken" rules about certain nouns in Spanish vs. English... for instance, both "agua" and "water" are almost always acceptable without an article; are mirrors just viewed (hah) differently than other concrete, quantified nouns in Spanish-speaking cultures?
That is an excellent question for which I do not have an answer, so I voted your question up.
Native speaker here, but not a grammarian or a linguist so this comes off the top of my head by analyzing how I and people I know talk. Also, I'm European, so this might be different for American Spanish.
In your example the article might make it "too" precise, as in "I don't have a mirror; I have two". In the same way when we don't have a computer we say "No tengo ordenador", not "no tengo un ordenador". It means roughly "There is not any computer among my posessions", and it's used especially when you don't have something that is supposed to be a fixture in almost every house, workplace, etc. With plural, quantified names, the lack of an article usually means either "I normally have some of those but I have run out of them", or "I never have those". Examples:
No tengo espejo - You have asked me if I carried a mirror in my purse but I don't have one.
No tengo espejos - There aren't any mirrors in my house.
No tengo televisor - There's not a TV in my house, for whatever reason.
No tengo cebollas - I looked at the fridge and noticed I have ran out of onions.
No tengo ninguna cebolla - Same as above (literally "I have no onions"), silightly emphasizing the fact that I have actually NO onions (maybe in response to a further inquiry about if I am sure about the non-existence of onions in my fridge ;).
Notice that when making a question the article can or can not be used depending on the level of precision and "proximity" you intend to convey. Examples:
¿Tienes un bolígrafo? - I'm not interested in whether you have ballpens in your house, in your desktop drawer or whatever, I'm interested in whether you have one on you right now.
¿Tienes DVD? - I'm not interested in how many DVD players you have at home, just if you have any.