No they didn't make mistake, the crusaders were the Rums and the west Europe were
engaged in the war, read
here
Yes, then the Latin Franks stabbed the Greek Romans in the back by keeping the the territories for themselves.
When the crusaders captured territory, these didn't become part of the Byzantine Empire. Do you believe the Muslims were not intelligent enough to understand this? I believe they quickly worked out the difference because they weren't stupid, perhaps you disagree?
Crusader states - Wikipedia
The Ottoman empire isn't Syria, isn't Egypt, isn't Iraq, but it's the Turks invading these areas,
the same thing with the Romanians(al rum) coming from today Europe and invaded some areas in the Mediterranean.
As I've probably told your 15 times to no avail, you keep thinking of modern nation states and modern concepts of nationality and race. You seem to think the Roman Empire was sort of like 19th/20th C European colonialism. Almost no one of any significance in the Byzantine Empire was from "Italy".
The Byzantine Empire was a Greek speaking Empire based in Anatolia. Anatolia, Syria, the Levant, etc. were Hellenistic (Greek) cultures, and populated by the same ethnic group as Greece was. From Greece to the Middle East (Greece, Turkey, Syria, Lebanon, Palestine, etc.) you had a common ethno-cultural grouping. This group was part of the same empire for 1000+ years. It wasn't a few independent nations colonised by Italians.
Empire was territory, and to rule territory you spread your culture and integrate locals into your society. In turn, your culture is affected by the culture of the locals so, over time, you create something new. This is what the Arabs did after their conquests. As you can see in the Middle East today, over 1000 years culture spreads a great deal.
Places in Asia/North Africa Graeco-Roman world for 1000 years were not imagining some glory days before they were colonised by Italians. They were Graeco-Roman societies the same as the ones in "Europe".
You think that because France, Germany and Britain were part of the Roman Empire for 400 years they are forever 'Roman', yet places that were far more important and were part of the Roman Empire for up to 1500 years were somehow independent entities colonised by Europeans and shouldn't be considered Roman at all.
This is a very stupid argument.