It is a central part of Christian theology that Jesus was sinless, which means that he kept the Torah perfectly, so if you think that he broke the Sabbath, then you should think that he was just another sinner who was in as much need of a Savior from his sin as the rest of us. Furthermore, the way that the Bible instructs us to tell that someone is a false prophet is if they teach God's people against obeying His commands (Deuteronomy 13:4-6), so if you think Jesus taught against obeying God's dietary laws, then you should consider him to be a false prophet. Likewise, it is a sin to add to or subtract from God's commands (Deuteronomy 4:2), so if you think he tried to subtract God's dietary laws, then you should consider him to be just another sinner who could not even save himself much less anyone else.
However, according to the Pharisees, if you didn't obey their traditions for how to keep the Sabbath, then you were breaking it, so while Jesus certainly broke their traditions, he never broke the Sabbath as it was intended to be followed. In regard to Mark 7, the topic of conversation was that you could be defiled by eating kosher food with unwashed hands and Jesus was simply denying that to be the case. His statement in Matthew 15:20 shows that at the end of the conversation he was still talking about not being defiled by eating with unwashed hands, so he never jumped topics from speaking against a man-made ritual purity law to speaking against obeying God. In
John 8:1-12, we have another example of Jesus following the law rather than breaking it. There was no judge to pronounce a sentence (
Deuteronomy 19:17-21), there was no man accused (
Leviticus 20:10), he didn't have any witnesses to examine (
Numbers 35:30,
Deuteronomy 19:5), and he did not have a confession, so if he had condemned her, then he would have acted in violation of the law. Just a few verses later Jesus said that he judged no one (
John 8:15) and he also said that he came not to judge (
John 12:47), so he did not exercise authority as a magistrate and did not condemn her, but he did recognize her action as sin, and told her to go and sin no more.