First of all, Jesus spoke Aramaic because of the Babylonians ruling judea in 587 BC. Therefore the language that Jesus spoke was the same as the language of Baghdad and Iran! Did Jesus regard himself as a Babylonian or as a roman or as a judean?
Second of all: why isnt the history (rise and fall) of Babylon not viewed as a big deal? I know nothing about it. Did they or their culture go extinct due to Islam?
Finally, I mentioned Iran was speaking Aramaic which is a Semitic language and not an indoneuropean one. We know this is correct because of the Ashoka rock inscriptions.
What was Babylonias relation with Persia, Rome, and it's acceptance of Christianity? Did they introduce any religions like the other major empires?
You seem very unclear about some basic history in this period.
Firstly, the Neo-Babylonian empire was defeated and absorbed into the Achaemenid Persian empire in 539 BC, so after this time it's not typical to talk about Babylon as an independent entity, though technically the Achaemenids would hold the title King of Babylon.
Aramaic was in widespread use throughout the Achaemenid Persian empire and likely became the daily language of the exilic Jewish community which later returned and 're-colonised' Judea.
Secondly, Aramaic was not the 'language' of Babylonia, it's a Semitic tongue that seems to have emerged out of the area surrounding Damascus, so speaking Aramaic was certainly not likely to cause Jesus to self-identify as 'Babylonian'. Especially since in the 1st century the cultural 'heirs' of Babylon would be the Parthian Empire. Jewish self- and national- identity was quite strong, and given that there's no reason to suspect that rural Galilean folk held Roman citizenship, Jesus probably self-identified as a Jew more than anything else.
For the last part of your question you are going to have to be much more specific about the time period you are referring to.