I am not sure it's entirely uncommon, at least in the UK. For example:
- My friend fell down a set of stairs.
It's more or less interchangeable with stairwell or flight but it's useful for setting the scene
- It is a large set of stairs
- It is a massive set of stairs
Where large suggests ha high, unknown number of stairs and massive suggests scale. They both imply the other to an extent. Flight, stairwell and staircase do not really work with that type of situation
The word Wohnung must have gotten more specialized over the years. My Sprach Brockhaus from 1951 lists Heim or Zimmer as a definition of Wohnung. My recollection from taking German in the 60's was that Wohnung was a general term for dwelling. Google Translate links dwelling to die Wohnung—apartment, flat, accommodation, property, place, dwelling
"Heim" is somewhat obsolete I'd say. Except perhaps when using "Mein Eigenheim" meaning "my (owned) home" or "my property".
"Zimmer" is specifically a single room. There could be a bathroom attached or access to a community bathroom elsewhere, a small kitchen corner could be integrated. The term "Zimmer" in your old book probably stems from post-war times. Because of limited housing availability whole families lived in single rooms back then. Today you'd probably find mostly college students living in a "Zimmer," either privately or in a dorm. I wouldn't call that a "Wohnung" though, whether after the war or nowadays.
The best translation of "Wohnung" is probably "condominium", but that's, of course, mostly an American term, known as "apartment" or "flat" in other parts of the English-speaking world.
So a "Wohnung" is mostly an apartment, a single unit in an apartment building or high-rise, i.e. there are other "Wohnungen" around it in the same building. I'd say a kitchen, living room, bathroom and at least one bedroom make a "Wohnung".
As soon as you have a "Wohnung" alone in a single building it becomes a "Haus". Even a "townhouse" (American) or "terraced house" (British), i.e. a house that is conjoined to other houses, either on one or two sides, but typically occupying the whole vertical space is already called a "Haus" (more specifically "Reihenhaus" (conjoined on two sides) or "Doppelhaus" (conjoined on only one side)), but, again, this is definitely not a "Wohnung" any more.
And, of course, a free-standing house would definitely not be called a "Wohnung".
They don't say that staircase is wrong.
They say residence is wrong, and say "home" is correct.
However, residence is a home, at least in English. A residential area, a residential establishment, etcetera.
I know you cannot merely change a structure of another language just to fit a different grammatical structure. But come on, residence is synonymous with home, in fact it's a more formal word for home!
It's simply a matter of singular vs. plural. "Stairs" is plural. "Treppe" is singular.
You say "I have a car" (singular), but "I have cars" (plural) - indefinite article is dropped when switching from singular to plural.
If you translate "Treppe" as "stairwell" or "staircase" to maintain the word being singular across both languages, the sentence translates one-to-one: "His apartment has a stairwell/staircase."
Likewise, if you wanted to say "His apartment has stairs" in German, a close translation would probably be "Seine Wohnung hat Treppenstufen" - both sentences have no article.