1. What role do arguments play in answering this question?
I think arguments have played an important role in analyzing and understanding the depth of this question, for mankind. Although the question itself seems factual (either it does or it doesn't), yet no arguments have been able to answer this question conclusively, despite many debates going on for centuries. One possible reason for that inconclusiveness may lie in our intuition and the way, humans define God and existence. We will explain why we don't think arguments conclusively answer this question, in more detail in answer to question 4 below.
2. Does Stroll offer any arguments that help answer this question?
Stroll himself doesn't propose any arguments himself, for or against, this question. He does analyze the two classical arguments for the existence of God, namely the ontological argument and the argument from design. The two arguments belong to sharply contrasting philosophical approaches namely rationalism and empiricism.
The ontological argument is based on rationalism and uses reasoning (much in the same way as mathematics) to make its case for the existence of God and is thus a priori. The ontological argument uses opposite contradiction to prove the existence of a being than which a greater cannot be thought. It proposes that since necessary existence or more strictly, non-contingent existence is greater than contingent existence, so God necessarily exists. This argument is put forth by philosophers' like Anselm, Descartes, Leibniz and Platinga who are all rationalists.
The design argument is rooted in empiricism and proposes that all matters of fact to be discussed based on experiment and observation (much in the same way as science). It uses observation of our world, complex interaction of forces of nature and the scientific laws governing them, to propose that this world is like a complex machine and hence must have a designer. This designer is God...