Your position, "God is not real because I cannot sense God as I sense the rest of reality," is quite well and boldly expressed. As an open question, it brings us as humans right to the point at which the gospel speaks. As a final conclusion, it blindly assumes victory after having slain only a straw man.
By "expressed openly" I mean within the limits of human finitude but without excluding the possibility of the objectivity and freedom of God beyond those limits. We can't intelligently say, "God doesn't exist because I can't touch him", while you could say that apart from Christ "God is not real to me because he has not made himself real for me within the limitations that my knowledge is necessarily bound to." For me as a human to know something or someone is real, that reality must present itself as both sensible and intelligible. Enter Jesus Christ.
The Gospel tells us that God, who cannot be touched or fully comprehended, has condescended to make himself known to us within the physical and intelligible limitations of human life and speech by becoming incarnated in Jesus Christ. God made himself knowable by presenting himself within our touchable and intelligible realm. However, even this is still not on the terms you describe. To touch Jesus' skin and hear his words was not to touch God directly or hear him directly, but to touch and hear that in which God had made himself fully present and through which he made himself known. In other words, one can take a position of doubt, saying Jesus was merely a man, his words merely human words. No overwhelming logical argument can fully refute this doubt. But for those, as Jesus said, "with ears to hear", God made and continues to make himself fully knowable in the sensible and intelligible reality of the man Jesus Christ.
"Did you just say 'continues to make'? Jesus isn't walking the earth today!" Yes, but God's taking human form and human speech in Jesus Christ as the Incarnation of his eternal Word has forged a new knowledge of himself in humanity that perpetuates itself through those who know it, the church. You can see and hear the church, understand its proclamation of the risen and everliving Christ; these are sense experiences you cannot deny having, it is just a question of your willingness to receive them as communicative of knowledge of God. How do we know God is communicating himself through the church's proclamation? You must approach God through them and see if he is there to be found. How? Through the means appropriate to him: prayer and worship.
This is the message of Christianity. Your fundamental argument, "God is not real because I cannot sense God as I sense the rest of reality," has met a counter claim. If you have made your argument "openly", you must consider the church's proclamation of Jesus Christ, God come among us in our sphere of observability, and make your judgement. As it is, your argument, if expressed in a closed way, is a rejection only of a straw man, something other than the Christian God who is defined by untouchability but has nevertheless taken on touchability for you and for me. This seems to be your complaint though: God is too untouchable in his eternal nature and too touchable in his human mediation. This is just complaining that God is too God.
At this point, the first two categories of tangible experiences of God you mentioned I would consider dealt with. Christians, having heard the voice of the eternal God in the proclamation of the gospel and thus having learned to correlate events and realities in this world, both fantastic and mundane, with their source and meaning in the eternal love and will of God for us in Christ, interpret both fortuitous synchronicity and pretty sunsets as the Creator speaking in his creation. Of course these things cannot prove the validity of this theological interpretation; that can only be validated by a prior encounter with God in his gospel. Of course there are other ostensibly plausible interpretations of these events. You've offered a common and compelling one, the one of the linear evolution of human understanding where we go from mythology, to religion, to naturalistic science. This story is forceful and persuasive, except for the fact that it offers no proof of itself. It is just as liable to the charge of total fabrication as is any meta-narrative of human or cosmic history. Just as the Christians' interpretation works perfectly well if God exists, so does yours if God doesn't exist. But this does nothing by way of offering any evidence for or against God's existence. Lets come back to that shortly.
Your point about Christians not believing in the God of the Bible I found totally compelling and convicting. You're right; most Christians don't really believe in the God of the Bible. The Bible presents us with something quite alien to our experiences, what Karl Barth called "a strange new world". Liberal Christianity explains all this away and repackages the Christian message as human progress, or social or personal enlightenment (God-lite). On the other side, fanatical Pentecostal and charismatic Christianity seems to deny the tension and convince themselves that they see pillars of fire and so on. Really, though, both are doing the same thing; reducing the Creator to the sphere of the created. The rest of us fall feebly somewhere in between. For myself, I seek to believe and be faithful to the God of the Bible, but I'm constantly faced with my incompetence in this regard; I can't help but feel a serious conflict between the world the Bible depicts, one in which God is seemingly ever present, active, and articulate, and my own world which feels much less spectacular. But this tension is necessary. If the Bible and the God whose revelation it mediates to me weren't so wholly other from the world of my experiences and expectations, it could not call me out of myself to a faith whose center is another Being entirely. It is a tension that seizes me, calls me to repentance for trying to resolve, and produces faith and hope in the man at the center of that tension. If the story of that man were pure mythology, I could easily dismiss it and there would be no tension. If it conformed to my experience and expectation, it would reveal nothing to me. This tension you have so well named here I will not apologize for leaving unresolved. God speaks in that tension and moves to resolve it himself in the fulfillment of history yet to come in Christ's return.
Let me finish by coming back to the issue of the equal footing of internally consistent worldviews. Both Christianity and atheism provide ways of looking at the world that accord fairly well within themselves. Their starting points, however, cannot be arrived at neutrally, but are bound up with the view of the world they provide. You have spoken of the experience of others from a worldview of unbelief; you answered Jesus' "who do they say that I am?" from a standpoint hostile to faith. Now he asks you "who do you say that I am?" This question cannot be answered neutrally. It calls your self and everything you think into question. You cannot evade the question with appeals to presuppositions about sense or evidence. If Christ confronts you, it is he and he alone on whom your answer must be based. Is Christ a liar? Is the proclamation and fellowship of his church a fraud? If you would answer yes, it cannot be because you can't touch God. You need to reject the actual message as it presents itself to you, not a caricature of it.