I know you probably do not like to hear that not every question has an answer, but, truly, not every question is answerable.
For example, is this a question, with an available answer? How is the why of why a what, if it is when?
Well, it has the form of a question, but no answer can even be assumed, much less answered.
No answer can be assumed, to the why of God‘s existence.
Does that mean there is no answer to why does God exist?
Well, we have no concepts, no words, no inkling to approaching the question.
For our tools, it is truly a conundrum, wrapped in a mystery, clouded within a puzzle.
We have no basis to even assume there is an answer, much less what it is.
It would probably be easier to answer the “question” of how is the why of why a what, if it is when.
Could God answer the question of why He exists?
Of course, but only He could reveal the answer : should there be one.
Your other question of what is God’s purpose is not clear.
How are you directing the use of the word: purpose? Purpose with respect to what, and in what manner?