The area that the Crusaders were focused on, Jerusalem and its surroundings, hadn't "belonged" to the Eastern Roman Empire for closing on 500 years.
The Byzantines had a much better case for some of the other territories, such as Antioch, the Crusaders glommed onto.
The Crusaders had sworn to return conquered land to Alexis, but rightly or wrongly they believed they were absolved from that obligation because Alexis didn't support them as he'd sworn to do, abandoning them to what he thought would be defeat at Antioch.
Correct and I agree with you. I only mention it because many try to use that as a reason or justification for the Crusades.
Pope Urban's reasoning revolved totally around expanding the influence of the western Church. Nobody in Constantinople asked for the crusades.