Gentiles are not the lost sheep of Israel.. What are you talking about?
My thread 'Who are the Gentiles' is available to read on my public view on this forum, I am sure it can answer all your questions.
The lost sheep of Israel are the scattered tribes across the nations.
That's right, they also became nations as well, fullfilling promises made to Abraham by God.
Gentiles were never known as sheep they were known as dogs.
I think your getting some key words mixed up, namely 'nations' and 'Gentiles', the Greek word
ethnos means 'one of the same tribe', of a like kind or class, from where we get the words 'genus', 'gentle', 'gentry', 'genteel', see the Latin GENS - meaning to beget or produce of the same kind - ethnos - see Liddel & Scott.
This lady was a gentile dog, not a gentile sheep.
Specifically a 'household dog', I gave you the word for this, it was common for Judah to see divorced Israel in this manner, however Jesus set the men of Judah straight on this saying in His ministry "I come only for the lost sheep of Israel' whom were scattered abroad, like this woman in Canaan land, who was a believer in the Messiah, just like the Samaritans were descended from Jacob, Jesus went to them early in His ministry because they were Israel.
It says it clearly right there, you just don't want to admit the fact your theology is flawed and does not match up with the bible because you are too prideful.
I know what it says and I'm explaining to you what it means, so you can understand, I don't wish that anyone dwell in darkness and ignorance.