If one means physical, emotional, and mental existence, as a discrete entity, as we believe a human exists, then the answer is no.
It is important to note that we don't necessarily know the full reality in which God would exist, just as we don't necessarily know the full reality in which we exist. It is also important to note that we ourselves are not perfectly discrete, and it is quite possible that other 'entities' may be less discrete than us--and also that we are less discrete than we believe.
This can lead to a lot of ideas on the forms of existence which God may have, though without existing in the same sense that we exist. But to a human being, the central way in which God does or does not exist is as a form of meaning. The concept of God gives a shape to that which we don't know or understand, and allows us to interact with it. Does it respond? Yes, in my experience. This concept of God allows us to know 'things' which we wouldn't otherwise know, in a different way than we ordinarily know 'things'. In this sense, God is the metaphysical, which, according to this conceptualization, has something which strikes us as a living and metaphysical quality. This quality may be understood to amount to a kind of existence.
Is God, in this sense, important? Or is God like some guy you pass on the street, who has no effect on your actual spiritual life--but a lot bigger or smaller? Well--are your friends important, is your mother important, are your teachers important, are you important, do these things have an effect on your actual spiritual life? Again, one's answer depends on one's definitions, one's goals, the nature of one's non-goals.
The conceptual nature of God is a different matter than the manifestations of God. But are certainly subject to one's own filtering and shaping, but in different ways. One of the keys to being able to have authentic spiritual experience throughout one's life is to understand that God does not have a limitation on permissible manifestations. There are patterns, but there are no rules. God may be singular, multiple, or simply diffuse. God may seem to manifest in full, in part, or through a representative. God may manifest in a moment, or in a whole identity. God may manifest in any form on the physical, emotional, and mental levels. And God's absence may also be a form of manifestation, whether as a total absence or a withdrawal and distant contact through messengers.
Meanwhile, what I have just written is my conception of God assembled through experience and reason and some inspiration. But there is never any guarantee of reality of any kind. A particularly idiosyncratic manifestation of God may be, in a crucial sense, more real than my well thought out conceptualization. And so on, with a kind of infinite regress.
An absolute understanding of God is beyond our abilities at this point, and it may well be that, were we to gain those abilities, we would find that in fact God does not exist in any ultimate sense--as a panorama no longer exists when you are in the middle of it. Many people have realized this. But that does not necessarily mean that God does not exist now, in relation to the abilities of perception that we now possess.
Unfortunately, many people simply can't get past the sense Old Man in the Sky. Rather than God as 'an ultimate metaphysical entity' (rather than the evasive 'ultimate' that contemporary theologians use, in order to stay away from anything controversial), they keep returning to this stamped on understanding, and then on the question of the existence of God they compulsively say "But there's no Old Man in the Sky!" At that point I'd say, throw monotheism out the window, and throw polytheism out there too if that has the same effect. Rather, consider the idea that the unknown or the metaphysical can and does interact with you.
After some time has passed, I think that it is important to bring back the sense of ultimacy, of power, that was traditionally associated with the patriarchal identity of God. It is more than fine if it isn't identity based at all. ...