According to the Trinity doctrine was the Son created by the Father? According to the Trinity doctrine was the Son created by the Father at some point?
Short answer: no he wasn't. That would be the heresy of Arianism.

In Trinitarian theology the Son is said to be "eternally begotten" by the Father. Admittedly this is conceptually sorta like being created by the father, with the additional caveat that there was never a time when the son did not exist (hence he is begotten "from eternity"), but metaphysically being begotten is completely different to being created because the son is not part of creation: he has always existed.

It's a hard topic to discuss though. Language starts to get very wonky and inexact when you start talking about transcendant stuff that fundamentally can't be talked about. God is ineffable after all

See the Nicene creed, which was formulated in the fourth century in order to condemn Arianism. It has the line 


  [I believe] in one Lord Jesus Christ, the only-begotten Son of God, begotten of the Father before all worlds (æons), Light of Light, very God of very God, begotten, not made, being of one substance with the Father;


Where it says "begotten, not made" you can read that as saying "begotten, not created".