1. Belief in the existence of a Creator and of providence
Well, I don't believe the Universe has existed eternally, nor do I think it likely that the whole complexity and highly ordered nature of our world was merely an accident. But on that same token, the idea that an eternal, intelligent superbeing wished it all into existence also strikes me as absurd.
What I can believe is that there is some creative force on which our world's existence depends on. That force may or may not have any sort of consciousness or intelligence or be a "person" in any sense, though I think it is more likely impersonal. Its "providence" can only be summed up as far as the qualities that the reasoned order of our universe supplied to life and living beings so prodigiously that they could survive and prosper through their own activities.
2. Belief in His unity
To term this force "His" is pushing it, but that's likely just metaphorical. Though I also have no problem in believing that there is only one fundamental creative/maintaining force on which the world depends.
3. Belief in His incorporeality
I also have no issue in understanding this force to be without material form.
4. Belief in His eternity
As far as the universe is inclusive to "all of time," since time itself is a product of our universe, this force would exist as long as the universe would - which would be "for all time." Essentially eternal since time is likely illusory anyway.
5. Belief that worship is due to Him alone
If there's anything that deserves worship it would be this force. Though I'm not thrilled with "worshiping" anything. I like to think more along the lines of feeling in awe and having deep respect for all that I see in the natural world. This is akin to the views of Einstein.
6. Belief that G-d communicates with man through prophecy
What is "prophecy" exactly? If it is understood in the sense that "prophecy" is man's connection with a new idea, where he might find a muse hiding in the regularity of everyday life, that I can understand. Since "God" is really the sum of all that is in existence, these new ideas and abilities to learn and to understand can only come from God. "Prophecy" is the point at which one can say, "Eureka!" or come to some deep emotional understanding where some connection is made through our mental powers which, along with all other things, from some creative force which underlies the whole. In this sense I can believe in prophecy.
7. Belief that Moses was the greatest of all the prophets
In following the Jewish tradition, Moses (assuming he existed, though even if he didn't and things were just attributed to him) was the one who set the basic foundations for what Judaism is and how it would be forevermore. He surely had great insights to produce such a solid basis from which Judaism has maintained itself for so many millennia and from which so many other people of the world through Christianity and Islam find meaning. In this sense, whether in history or in legend, Moses was the greatest prophet.
8. Belief in the revelation of the Torah to Moses at Sinai
What does it mean to be a revelation? Ultimately, all things come from this creative force so how can the Torah not be? Does this mean it is divine in origin? Not really. But we must hold strong meaning for it anyway because we have given it meaning through so many years of tradition and history that now the Torah is (and has been) an integral part of Judaism and what it means to be a Jew.
Do I think a personal, intelligent superbeing gave Moses the Torah word for word? No. But I do think Moses (or whoever wrote it and later attributed to Moses) hit on some great ideas and in this sense we ought to hold it in high regard. Especially since the Torah has such deep meaning in Judaism and to the Jewish understanding of identity that it cannot really be separated from what it is to be a Jew.
9. Belief in the unchangeable nature of the revealed Law
Even the Mishnah and Talmud don't really hold by this as they create so many new ideas and sometimes controvert the basic meaning of the scriptural text. What is a prozbul exactly? And how come an "eye for an eye" isn't taken literally? We can thank our forward-thinking Jewish ancestors for that. I think Halachah is of vital importance to Jewish identity and Judaism's longevity, but that it is of divine origin or forever unchangeable, that I do not agree with.
The basic foundations of the Law, however, respect for human life, compassion for your fellow human beings, maintaining the environment, etc. These I do see as incontrovertible and fundamental. The technical laws themselves can and should change as the social situations change (for example how women are viewed and treated in Orthodoxy today) but the fundamental _spirit_ of the laws must be maintained.
10. Belief that G-d is omniscient
Since I don't believe that "God" is even conscious or capable of "knowing" anything, it makes it incredibly difficult to imagine this apersonal force being omniscient, i.e. all knowing. Though, again, if all things are inclusive within "God" then the sum of all that is known must be what God knows. Thus God knows all that is known - all knowing.
11. Belief in divine reward & retribution in this world and in the hereafter
Perhaps in the sense of strict rationality, one can find that generally you will live a happier and healthier life if you are a good person, are friendly and cooperative with others, etc. People who are mean and uncooperative will often live poor lonely lives. There is some balance here. If you are a rotten person, people will generally not want to be your friend. Also, if you break the law, perhaps you'll get away with it, but many other times you will get caught. The world is not perfect and there are countless examples of injustice, but there is also some justice to be found. I believe it is humanity's responsibility to bring as much justice as can be brought into the world.
In the sense that humanity is derived from the creative force and concepts like justice and right and wrong are also from this source, reward and punishment become possible in human experience. Were the universe mindlessly random, these concepts would have no meaning. These things do exist, but I think only to the extent that human beings bring them into existence.
In the "afterlife," I can only say that justice is done on the individual through how he is remembered by the people that knew him and what kind of reputation he held. Do you wish to be remembered as an asshole or a great contributor to the betterment of humanity? That choice is up to you. I don't think, however, that really any part of us as a conscious mind survives the body in death. What survives us is our works, our name, and our reputation.
12. Belief in the coming of the Messiah
I believe that humanity can improve itself. That we can all be better people. That we can invent new technologies to make our quality of life better. That we can discover new philosophies and new ideas with which to give our lives more meaning. In this sense I believe that we, as a species of moral, intelligent consciousnesses, are capable of progress on all fronts and may one day reach a "Messianic" age where all of humanity lives in peace with one another and where each individual's life is free from fear, hunger, pain, and need.
I don't believe that one man will ride up on a donkey and then see to the conquest of the whole world which will lead to Israel's ascendency on the world stage. As Gersonides once said, "a peace that comes from fear and not from the heart is the opposite of peace." I do think that one day the world will find peace through mutual respect and brotherhood and not through fear of the sword.
13. Belief in the resurrection of the dead
This "ikkar" has always confused scholars. The afterlife, in Judaism, is never the central feature of any theological theory and really holds little place in general Jewish metaphysics. Although it was standard fare to believe in an afterlife, what that afterlife would be like was freely speculated on and to this day there is no "Orthodox" view on the afterlife. Why then did Maimonides focus on this and make it a _fundamental_ of the Jewish faith?
I don't think resurrection of the dead is going to happen. And although you may read possible science fiction considerations of such a human project to reanimate all the dead human consciousnesses in the future (e.g. 3001 The Final Odyssey), I just don't think it is very likely or even possible. I think we get one shot at life and there is no "reset" button. Let's make the best of what we've got.