Halo Daniel,
Yes. As a person who was brought up Catholic and did not have any knowledge about Islam and Quran, I had been trying to let go of my ignorance and try to learn the Quran and the historical background of Prophet Muhammad ministry on earth.
It is clear that the Quran was revealed for mankind, but the first main target of Quran to establish Islam is the immediate surrounding of Muhammad. So, yes, if one want to deeply understand how Islam related to the Jews and Christian in 6th century Arabian, one must put it in context; how was the nature of the Jews and Christian at that time, in that place. Which I think we both agree, that the Christian in Muhammad surrounding was not Western Church Christian, but rather 'Easterners' and 'Heretical'.
The thing is, probably, Islam and Christianity has separated so widely, that they seem to be two conflicting forces nowadays. Jesus followers have been conditioned that Islam hates Jesus and misrepresenting Jesus, etc. Islam society has also been conditioned to think that Bible is highly corrupted and that Christian are crazy for worshiping three gods, which of course, in Trinity point of view, it's not like that. What should happen is communication that excludes all those negatives preconception, and we should humbly seek the agreements and understandings from all the revealed words from God, which of course as confirmed by the Quran: Torah, Psalms, and Gospels.
It's nice to know you Daniel, let's pray the best for our nation and mankind.
Salam