Ah another age old and still unanswered question. Yes those are both from the same book. Old Testament had stonings. Sounds harsh I know. I have heard of 2 reasons for the change.

1. Having a kid can change your point of view

and ( I think more likely)

2. The purpose of all the laws in the Old Testament was to show us how impossible it is to be truly righteous before God, hece the need for a savior.

I would also like to address the misconception about judging. The Bible clearly states "not" to judge, many times. Clearly the subject is vast and can be confusing to try and understand all the complexities.

Lastly, on your final question, I would say because it advocates good things. There is a point in the Bible where somebody asks Jesus, "what is the great commandment?". Jesus said, if you read the whole Old Testament and you get this, to love God and love people, then that is a pretty good reading. The book also says to take care of the poor, love your enemies, and have a relationship with God. Sounds like a pretty reasonable book to me, certainly could be words of God.

Sounds like u might as well get used to it.

If the world hates you, keep in mind that it hated me first.
If you belonged to the world, it would love you as its own. As it is, you do not belong to the world, but I have chosen you out of the world. That is why the world hates you.
John 15:17-20

Matthew 5:12
Rejoice and be glad, because great is your reward in heaven, for in the same way they persecuted the prophets who were before you.
Matthew 5:11-13

Even when someone rejoices in the Savior, they are still sinners, once again, there is no judgmental label put on it.

Everybody sins, but that doesn't mean that I label them good or bad. It's kind of a fact. My dog needs to eat food, but I don't attach any judgment to that. It's just the way it is.