(1) as a FULL sentence would be :
"We are not allowed THE listening to music in this building. "
So, 'listening' is a gerund in your sentence, and so a NOUN.
Now - what may be confusing you is reading a sentence such as this:
"On this diet, we are not allowed ice cream."
Here, 'ice cream' is a noun, just as in (1); but the FULL sentence would be:
"On this diet, we are not allowed TO EAT ice cream."
The meaning of 'allow' in your sentence:
allow : let (someone) DO something ( that is, an ACTION, as in 'to listen' and 'to eat')
That is why, whenever we use 'allow' in sentences in the Active Voice, such as yours, we always use the infinitive, not the gerund.
compare Passive Voice:
"Listening to rock music is not allowed in this convent school."
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