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The home could have been burned in the fire if the firefighters hadn't arrived.
My intuition is that although the form is similar to third conditional, it is not third conditional. Because we use third conditional about wishes in the past. Also this could be written in following form using just "could have" and not "could have been" Could have burned.
This is a form of 3rd conditional (which combines "If + past perfect/past modal, perfect conditional - note, your example has the clauses reversed).
It is a bit more difficult for a language learner since the if clause comes in the second part. Also adding to the difficulty is the use of "could have" which adds an extra layer of "imaginary situation".
Let's see: 3rd conditionals express the opposite of reality (imaginary situations, wishes, desires etc)
If the firefighters hadn't arrived
=
The firefighters arrived
And
The home could have burned
=
The home didn't burn
Note that "could have burned" indicates that there was only a chance for it to burn if the firefighters hadn't come.
I think the sentence sounds better (second part of your question) if we say:
The home could have burned (down) in the fire if the firefighters hadn't arrived.
"Been burned" as a passive structure doesn't sound quite as good to me as the active structure I've suggested here.
