I'm not sure where the title "God the Son" is in scripture. But I think generically it's understood to be referring to the second person of the Paradise Trinity which consists of God the Father, God the Son and God the Spirit.
Yes, certainly because it makes total sense that the person who created the universe and all of us humans who live in it, would eventually desire to experience living the life we live exactly like we do.
"Jesus was God and man—always and even forevermore. And this God and this man were, and now...
God the Father and God the Son are not the same person. Just like with all fathers and sons.
The Father and the Son both bear witness because they both dwell within us.
"The kingdom of God is within you."
Perhaps since Jesus was the incarnate Creator, he wanted to emphasize that he was living a life that was 100% human. Experiencing everything that each and every one of us must experience completely. From beginning to end.