Indeed.
He said that the Sabbath is made for man. The Sabbath is a previous gift from God that was made for our good and was not intended to bring about situations that are to our detriment.
No.
The issue is that some of God's laws appear to conflict with each other such as with God commanding to rest on the Sabbath while also commanding priests to make offerings on the Sabbath (Numbers 28:9-10), however, it wasn't the case that priests were forced to sin by disobeying one of God's commands no matter what they chose to do but that the lesser command was not intended to be understood as preventing the greater command from being obeyed. This is why Jesus said in Matthew 12:5-7 that priests who did their duties on the Sabbath were held innocent, why David and his men were held innocent, and why he defended his disciples as being innocent. This is also why it is lawful to get an ox out of a ditch on the Sabbath, why it is lawful to circumcise a baby on the 8th day if it happens to fall on the Sabbath, and so forth.
Some Pharisees had reasoned that it is unlawful to work on the Sabbath and that healing is work, therefore it is unlawful to heal on the Sabbath, however, we are also commanded to love our neighbor as ourselves, we would not be doing that if we refused to heal them, and no command was intended to be understood as preventing us from obeying the greatest two commandments, which is why it was lawful for Jesus to heal on the Sabbath. So there are some forms of work that are permitted on the Sabbath.
He didn't. In Deuteronomy 4:2, it is a sin to add to or subtract from the Law of Moses, so the position that Jesus changed it is the position that he sinned and is therefore not our Savior.
Jesus did not, but rather what he taught about divorce is in accordance with the OT, which does not permit divorce for any reason.
In Matthew 4:4, Jesus said that man shall not live by bread alone but by every word that comes from the mouth of God and he notably did not specify that he was only speaking about a subset of moral laws. Morality is in regard to what we ought to do and we ought to obey God's character traits, so everything in the Law of Moses is inherently a moral law. The Son is the radiance of God's glory and the exact likeness of His character (Hebrews 1:3), which he embodied through his works by setting a sinless example for us to follow of how to walk in obedience to the Law of Moses and we are told to follow his example (1 Peter 2:21-22) and that those who are in Christ are obligated to walk in the same way that he walked (1 John 2:6).
He wasn't.
The Law of Moses is perfect (Psalms 19:7), so it has no room for improvement.
We embody what we believe to be true about God through our works, such as with James 2:18 saying that he would show his faith through is works. In other words, the way to believe in God is by embodying His character traits, so the way to believe that God is compassionate is by being compassionate (Luke 6:36), the way to believe that God is holy is by being a doer of His instructions for how to be holy as He is holy (1 Peter 1:16), and so forth, so if someone refuses to follow God's instructions for how to be holy as He is holy, then they believe in a God who is not holy, which is an incomplete understanding of the God of Israel, who is holy. Instructions for how to embody God's character traits can't be changed without changing God's character traits, but God is the same yesterday, today, and forever.
In Deuteronomy 13, the way that God instructed to determine that someone is a false prophet who is not speaking for Him is if they speak against obeying the Law of Moses, so it is either incorrect to interpret Jesus as doing that or he was a false prophet. Christians teaching wrongly that Jesus spoke against obeying the Law of Moses is sadly one of the biggest reasons why Jews have rejected him as the Messiah, and if Jesus had done that, then they would have been correctly acting in accordance with what God has commanded out of love for Him. To follow a different set laws for how to embody a different set of character traits would be to not follow the God of Abraham, Isaac, and Jacob, who has the character traits that the Law of Moses was given to teach us how to embody, which is why Deuteronomy associates leading people away from following the Law of Moses with leading people to follow other gods.
Jesus spent his ministry teaching his followers to obey the Law of Moses by word and by example and the reason why he established the New Covenant was not in order to nullify anything that he spent his ministry teaching or so that we could continue to have the same lawlessness that caused the New Covenant to be needed in the first place, but rather the New Covenant still involves following the Law of Moses (Hebrews 8:10, Ezekiel 36:26-27).
Jesus was having a conversation in regard to a tradition of the elders in regard to whether someone could become common by eating bread with unwashed hands, so he was not even speaking about eating animals. Jesus had just finished criticizing Pharisees as being hypocrites for setting aside the commandments of God in order to establish their own traditions, so he should not be interpreted as turning around and even more hypocritically doing what he just finished criticizing them as being hypocrites for doing. We should be careful not to take things that were only said in regard to the traditions of men and apply them as if they were said in regard to the commandments of God.