itinerant,
regarding your question.....
To the Jews, Gentiles were dogs -- outcasts -- becos they did not worhsip the God of Abraham, Isaac and Jacob and becos of their hedonistic lifestyle, idol worship etc.(But actually, the word in Mark 7:27 is less harsh and means puppy rather than dog.)
Basically, the time for preaching to the Gentiles had not yet come (it came with Paul commissioned by the risen Lord to preach to the Gentiles). In Bible principle, its always Jew first than Gentile. eg in spreading the Gospel, even in giving/offering. that's why NCC believes in offerings to Messianic (Christian) Jews if we want to hasten the spread of the Gospel worldwide.
The bread refers to the Word or actually healing/deliverance (eg when we take communion the bread is for our health). So Jesus was telling her (I believe he was testing her faith) that it was not time for him to minister to a Gentile Greek woman like her yet. The children's (Jews) bread (Gospel, healing etc) should be given to them (Jews) first then followed by Gentiles.
But she reached out in faith and took her rightful 'place' as "dog"/gentile (And she answered and said unto him, Yes, Lord: yet the dogs under the table eat of the children's crumbs.)
You see, she came to Jesus 'as Jew', Jesus corrected her, she took her rightful place as Gentile, then Jesus cld reach out and deliver her daughter.
Sometimes we need to take 'our place' and stop trying to be somebody we're not, b4 God can reach out. eg Lord I can't do anything to earn healing, i just receive by faith thru grace.
That's the jist of it from my pastor