Do they actually delete the files? Or do they just remove public access to them, but keep them on their servers?

Maybe they send the data to law enforcement automatically? (I've heard that they do that for child pornography; maybe they do, or could do, the same for terrorist material as well.)

This.

It's naive to think these videos are deleted. Homeland Security shows up and YouTube has no problems giving them the videos. (They probably have an entirely separate search interface that does nothing but surface that extremist content for the Feds. So Homeland Security probably doesn't even have to leave their office.)

Only if Google has a branch in Syria that is destroying the evidence for a legal action in Syria. Laws are linked to territories, US law is not applicable in Syria and the other way around.

> Laws are linked to territories, US law is not applicable in Syria and the other way around.

Doesn’t stop countries from trying. The CLOUD Act in the US is the US trying to extend its influence into other countries.