I apologized for being rude to Paul.
Being kind is a virtue.
His being dishonest did not help him socially.
These phrases are clearly functioning as nouns but the participle (being) seems to be functioning more like a verb than a noun. Are these noun phrases or verb phrases functioning as nouns? And how should being be classified as word (PoS)? Is it a participle, a verb or a gerund? This is really just a question about syntax, I understand what these phrases mean. I'm more interested in the terms of analysis.