Firstly I agree with you, it is not obvious or immediately evident that reality bears any semblance to the divine. In an World of imperfection it is natural to wonder whether there exists a God behind it all. It is even harder to imagine sometimes that God may be immanent within this reality.
But when we study the Worlds religions and learn something about the Nature of God then we start to see some correspondences. For instance, on a very simple level, that which is God in the various religions is almost invariably described as having both a male and female aspect i.e. the Shiva/Shakti of Hinduism, the Yin & Yang of Taoism and the chapter Genesis in the Bible states that 'In the image of God he made them, Male and Female he made them.', which suggests that the image of God is both Male and Female. So correspondingly in reality we encounter in life male forms, female forms and every androgenous permutation in between. Even at the level of sub atomic particles everything is paired in what can be imagined as male and female couplings, with the exception of the photon or light which would correspond with God in its most timeless and indescribable form.
Other descriptions of the divine nature that are found throughout the mystical literature of the Worlds religions such as the Cosmic tree, the process of Emmanation from a point and a return to a Source also find their correspondent manifestations in forms and processes of
the physical illusory Universe. There are many examples of metaphysical descriptions of God found in the Worlds great faith traditions which seem to have their correspondent counter parts manifested in this virtual reality of projections.
It is as if God has a Cosmic image which is also a dynamic process and this image of God is manifest at all scales and all places in the Universe but in a slightly distorted or imperfect manner. If we imagine God as a template, then from this template all reality is stamped out and created. However at the same time the impressions made by this Cosmic template undergo a series of what mathematicians would call affine transformations, which can make the original template hard to make out in the image or projection created. Fractals are a good metaphor for God in this respect in that Fractals are objects where an overall geometric form is repeated endlessly within itself at all scales but in a slightly altered manner.
When we see things from this perspective then we start to see God everywhere, in the everyday, the mundane, the magnificent and even in the most dark, imperfect and evil of situations. And of course when we apply this principle to human beings then we get a glimpse at something divine in human nature when it is at its best and most evolved. Of course we are imperfect and life is filled with tragedy, traversity, farce, irony and frustration. But we may strive to better ourselves and indeed even aim for perfection. In the process even if we don't become perfect, still in our lives we drive forward the process of society and the unfolding of life. But every once in a while, in certain special moments in a person's life and in certain special lives or at certain points in World history, perfection emerges and it is then that the divine really does become fully apparent in this reality.
In response to the assertion that projections in this virtual reality cannot be extensions of the divine the answer is yes they can be. This relates to the puzzle of how the many, i.e. you, me and every consciousness in the Universe from the beginning to the end; can also at the same time be one and undivided i.e. God. This is a recurring idea in the mystical literature and is found in main works like the Bible, Koran and Bhagavad Gita. The answer emerges when we realize that another Universal and sometimes hidden truth behind all World religion namely reincarnation, goes hand in hand with the idea of the God within or that everyone is God. The truth is that the one is also the many and yet undivided by being the many, one life at a time. So in the same way we as humans sleep and awaken to a new day then so it is that as God we die and awaken to a new life. The projections that you see around you are indeed mere manifestions of your consciousness that is God and are existent only in your awareness, however these projections are also your past and your future lives. They are past and future manifestations of God as you yourself are. O.K. I know it's not so easy to understand when communicated like this in a single paragraph. I'll be writing more about this on the main website with diagrams to aid explanation, sometime in the future.
Finally in response to your question, 'How does one live in a World in which everything is an illusion?', Well once you realize that everything is illusory then life becomes a quest for the real, which is God and which is your true nature. All things in this illusory reality then become only meaningful in relation to the quest for awakening or God realization. To continue with our modern metaphor, this is the point when the virtual reality simulation pauses for a moment and we realize that we ourselves are the virtual reality simulator. This state is variously called The Kingdom of Heaven, Moksha, Enlightenment, Return to the Source etc. etc. etc. I'm going on a bit, I'll end here.