My answer to you question Do we worship the same God... If you look at the history I would say it completely supports the fact that we do. When Muhammed (pbuh), revealed the message of the Quran the intention was to bring pagans into monotheism,disbelievers into belief, Jews back to the rememberence of what had been given to them before and Christians into pure monotheism (without ascribing partners to God like saints, clergy and the worship of prophets) and to remove some of the new practices that were slowly finding their way into Gods message. Some accepted, believing God, the same God, had revealed His message again to the promised one spoken of by Jesus(pbuh). Others rejected that Muhommed was a prophet, but didn't necessarily think he was proclaiming to speak for a different God.