I guess he doesn't need to specify targets, when folks can just fill in the names in their own minds. I doubt he would hit one "cultural" target. I find difficult to understand when people who say Trump is constantly lying suddenly decide they think he's truthful.
People lie. Politicians lie even more. I don't expect better. As for your questions? I only saw one that I didn't bother addressing because the premise is weak. I think he's bluffing on destroying cultural sites, he just doesn't want them hiding there.
I guess it depends on what is seen as a cultural target. For instance, in the US hitting a pride parade might be seen by a foreigner as being of greater value than the National Archive...(We don't pay that much attention to the Constitution in Government now, so it can't be that important, but we seem to kowtow to every whim the alphabet folks have.)
So, pre-Islam culture might be seen as something which would not affect the current population in power, there, and nothing to target. It is a question of cultural relevance to the people alive there and in power, whether that target might have impact, or merely cause worldwide outrage. The former is generally desirable, the latter to be avoided.