What is the president's role in the legislative process?
He can veto bills and can issue an executive order.
What components of the dynamic legislative process is mostly left out of the textbook legislative process?
Constitutionally speaking the US President plays no role in the legislative process other than what influence he may exert on certain legislation by virtue of being the leader of his political party. Only when a bill is passed by Congress an is sent to his desk for signature into law, does the President affect any law.