Learn Swedish: Jag behöver lugn och ro.

Yes, it is a very common Swedish idiom that matches the English "peace and quiet" perfectly. While lugn means "calm" and ro means "tranquility," they are almost always paired together in this exact order when you want to express a need to be undisturbed.

I learned lugn as an adjective meaning "calm." Is it a noun here?

Yes, here it is the noun ett lugn (a calm or peace). It is paired with another noun, en ro (a rest or peace). Many Swedish words can function as both adjectives and nouns without changing their spelling.

Why isn't there an article like ett or en before these words?

Just like in English, you do not use articles with abstract, uncountable concepts when speaking generally. You say "peace and quiet" rather than "a peace and a quiet," and Swedish does the exact same thing here.

Do people ever use the word ro completely on its own?

It is possible, but fairly rare in modern everyday Swedish outside of poetry. You will mostly encounter ro in this specific fixed phrase, or inside related words like oro (worry or anxiety) and the verb att roa (to amuse).