Who - or what - exactly is he? What type of thing is he?
No one knows. It is also usually claimed that we have no way of understanding it as it exceeds our brain abilities.
Is he even a "he" - is he a person or something else?
It is not a he.
It is not a character.
It is something that cannot really be explained or understood.
It governs the way things work in our existence.
How does vacuum look like?
how does light look like?
We only see a fraction of what we KNOW is out there.
This is a hard question. I would argue its the number one question humans try to answer.
The Jewish belief claims it want us to experience itself.
The same as life cares about surviving.
Depends on what you mean "mind", but I would argue it does.
Does he have a personality - and if so what is he like?
Not really.
Its like asking of humans have a personality.
Does the universe have a personality?
What has he ever done for us?
Everything.
Who am I in relation to God?
A small fraction on a whole.
This fraction as being a part of a whole, is no less or more important the the whole itself.
Who are you in relation to God?
The same answer.
It what drives everything in motion and keeps it all as it should.
What does he want us to do?
succeed as an individual, community, specie, life, existence.
So it seems
Not really relevant.
It comes from something that cannot be explained.
Does he favour some people over others? Or some groups over other groups?
Not really
it will be very contradicting to its own logic.
What is he going to do in the future?
Only god know
The Jewish belief suggests it will not allow humans to become extinct and in the most desperate time of humanity, it will help and direct us in the right direction (actually its much more complex than that but that is the general idea).
Is it possible to have a relationship with him?
I think you are in a relationship with it whether you want it or not.
It will be the same as asking if it is possible for a fish to be in relationship with the sea.