Chapter 7 begins with the continuing conflict between Jesus and the "religious" folks. The whole hand washing, bowl washing discussion has nothing to do with sanitation -- first century people did not know about germs. This has everything to do with ritual cleansing. This ritual cleansing evolved over the years to the place where the tradition of doing it became of prime importance. This is another example of the Rabbis building a "hedge around the Torah". These traditions evolved to protect people from actually getting close to breaking the law. If the law said "no work on the Sabbath" the "hedge" defined exactly what work was. And so forth. This is legalism at its worse -- many many Christian groups fall into the same trap. When the traditions and the external actions become more important than the actual relationship with God and other people we have fallen into legalism. Note in this passage that Jesus clearly says you are not made sinful or unclean by what you eat "it is not what goes into you that defiles you". What is sinful or unclean is what comes out of our own brokenness -- from our broken selves comes murders, greed, pride, slander, adultery, etc. Change the inside and the outside changes. Focus only on the outside and, in the words of Jesus, we simply become "white washed sepulchers (tombs). Early Christians understood this to mean that the Old Testament dietary laws are no longer valid -- this understanding was reinforced by Peter's vision in Acts 10.
We learn in the story of the Syrophoenician woman that Jesus came, initially, for the lost house of Israel. But he extends here and in several other situations the new found grace to those outside the family. The "dogs" conversation seems harsh in the English -- the word for dog that both Jesus and the woman uses in this story is in the diminutive form and could be translated "puppy". The exchange is more playful than exclusionary.
There is a ministry to the deaf in Syracuse that calls itself "Ephphatha (be opened)" based on Mark 7:34. Ephphatha, (be opened) Talitha cum (little girl arise) (5:41) Eloi Eloi lema sabach thani (my God my God why have you forsake me -- a quotation from Psalm 22) (15:34) and Marana tha (Lord Come) are words that the Greek New Testament preserves in the original Aramaic. Aramaic would have been the dialect of the common people in Palestine of Jesus time. Hebrew was the language of worship and faith, Greek the language of commerce and education, Latin the language of the conquerors -- people had to know a little of several languages to navigate the world in which they lived. Why were they preserved in the Aramaic? The early church may have seen them as "words of power" or may have kept them because they were difficult to translate or they may have been preserved for other reasons (find 5 biblical scholars and get 7 opinions on the subject). (grin)