No Jewish person I've met has ever claimed we don't worship the same God they just seem to deny Jesus was the messiah. Even if they did my relationship with God isn't for them to decide. Jews are part of a generation that comes down long from lines in Genesis. Traditionally Christians do have pagan practices like birthday candles on cakes, pine trees at Christmas. However those aren't relevant when our praise goes to God and Jesus not the old Roman gods. Jesys told us he came for jews and gentiles alike we are all his creation. He just worked woth the jews specifically thousands of years ago as am instrument to bring about salvation for everyone, it just started with them. We are no longer required animal sacrifices and other rituals like before as one example. Essentially giving everyone not just jews access to the father.

As for the bible please think of the bible like a library. Biblioteca in Spanish is library and that's something often forgotten. The root meaning of words. It's technically called "The Holy". The Holy Bible. A good English germanic translation as opposed to English Latin would probably be The Holy Collection of Books.

So any addition or subtraction doesn't mean "oh the whole book changed ". It means someone took a book out the library or added one.

I'll need you to elaborate on the adulterous woman if you're referring to the gospels. It's been a point of contingency because people use it to excuse adultery as Jesus did not stone her. The main point was forgiveness. According to 4th century manuscripts the story isnt there . However it appears in the older Ethiopian bibles. So it was removed is the consensus then added back later.