To answer your daughter's last question first. OT was written by and for the Israelites, the Jews. God wanted them as a people for Himself, and still does, in spite of several thousand years of history, they are still His chosen people, but that is another story. They were given a list of commandments to live by, and if they broke them, they were to make sacrifices to atone for their sin. These sacrifices couldn't stop them sinning though.
When Jesus came, he changed everything. When He lived, he showed us how we should live, a pure and sinless life, but it is s till not possible for us to do that, we all sin and fall short of the glory of God. When Jesus was crucified, he became the sacrifice for our sin, he paid the price, and we do not have to. But to receive that sacrifice we have to acknowledge Jesus as our Lord and Saviour.
If we don't then the wages of sin are still the same - death. Through the death of Jesus we have everlasting life, he comes to live in us by the power of the Holy Spirit, the Holy Spirit is our comforter and guide, and a help in trouble.
Now to the homosexuality bit, which exercises so many people. God has called us all to be fruitful and multiply. Homosexual relationships, same gender sex, cannot do this, and as such are worthless in God's eyes. The difference between OT and NT is that old testament laws asked for stoning, in the NT Jesus says love one another as I have loved you. Another stoneable offence was adultery, but when Jesus met the women caught in adultery, he said her sins were forgiven, to go and sin no more. God has not changed, he still calls homosexuality an abomination, but we as believers we are called to hate the sin, but love the sinner.
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