With all due respect to you, there are some questions that are invalid questions that don't exactly have to be asked. There is no "have to" with God. God is his own authority - he cannot be subject to another being because otherwise he would not be God, and that being would be. There is no have to, and only want to, with God. God only does what he wants to do.
He wants us to exist, and thus we exist. Asking why we "have to" exist is like asking a kid why he or she has to have a toy. Clearly, the child doesn't truly need the toy, but he or she wants it. Likewise, God doesn't need us, but he wants us. However, that is not truly a full picture of God, because creation is also an expression of God's nature as a work of art.
Desire also defies reason, as does self-expression and artwork. 
Since those are God's motivations for creating us, our existence defies reason. You are correct.
Reason is a tool for understanding reality and deciding the best course of action for a group of human beings at a certain point in time. Its biggest fallacy is that its conclusions frequently lead human beings to doing things that benefit themselves at the expense of other human beings. This leads to violence. The Enlightenment has a lot of atrocities to answer for, despite the fact that it was better at spreading the Gospel than all of the ages before it.
It doesn't. Time cannot begin or end, because it defines beginnings and endings.
It also cannot loop in and of itself, because the scientific definition of time is based on entropy. Without energy moving into a more useless form, the universe would be static and nothing would ever happen. Sitting in this chair, my body is using up energy to move my fingertips to type this post, and my computer is using energy from the wall outlet to display it for my review and send it to you. If my body wasn't using/degrading energy, my fingers wouldn't be able to even move.
(My brain uses energy to think, so entropy is required for conscious thought. So without entropy, there would be no human reason. )
The best you'd be able to get out of time is a death spiral of lost energy, unless there was input from an infinite energy source to input into time to reset the loop. That infinite energy source is likely to have a consciousness, because it has infinite energy, so why not have one?
Also, if the universe was in fact run by an infinite energy source with no brain, the infinite energy would apply to the universe on an essentially random basis. I would have a 50/50 chance of a box of Cheerios appearing on my doorstep, or a Rolls Royce, or a catapult scorpion. Further, I would have a 50/50 chance of not even having a doorstep, or my house turning into an carriage. The universe would be total chaos. That is not our universe, which is highly ordered by very intricate natural laws that, among other things, indicate the odds of a catapult scorpion appearance to be much less than 50/50. (The fact that catapult scorpions do naturally not exist may be the foremost factor in this calculation.)
But in said chaotic universe, time would not really exist. In order for time to exist, it must be the product of an infinite energy source, because time is an an infinite degradation of energy (entropy) that cannot begin or end, and that energy source must have a consciousness to regulate itself so that it doesn't mix its own energy with the degrading system.
This universe requires God as a precondition of it existing in its current form. An infinite being has infinite action potential, so His desire for us to exist is infinite. This is already a basis for the idea of God's love. We know that goes much deeper through the actual revelation He has given us.
We also know that there aren't multiple gods either - those don't have to exist, and they don't - because no amount of finite energy makes infinite energy. You have to finite the energy to have a fight. And no infinite energy, no time and no universe - the infinite being with all of the infinite power would be in command, and since he has infinite power, he has infinite ability to defend us from other infinite beings, and He is the only one we need to deal with. So we can throw out the idea of multiple gods. Meanwhile, we can also throw out the idea of God as tyrant. He is the ultimate authority, but He greatly desires us to exist and doesn't need to have a fight, because He can just win.
However, He would want us to recognize his Authority and Infinite Power, because as finite beings picking a fight with Him is a battle we can never win. And He has an infinite desire for us to exist, not destroy ourselves by fighting with Him. However, we are finite beings and struggle to comprehend this reality, but He wants us to understand Him for this reason (among other reasons that he has revealed in His Word).
Now why God decided not to make spacetime circular is unknown, but the thing is, if you make time circular, our current entropic brains would likely reset every time time reset because the energy would increase to a previous state. If God wanted us to know who he was, him intervening to reset time (which he would have to do at intervals to resupply the universes' energy and thus reset time) would be counterproductive. There are also numerous other problems having to do with sin and punishment and memory that this would produce. For one thing, if time reset regularly, we wouldn't die.
I'm not sure I understand this post, but I think you're referring to what reality will likely be like in the New Jerusalem after the demise of this current universe. That universe has a time system that doesn't run on entropy, so you would have a spiral of infinite time resets with our brains somehow retaining information each time the system reset. This would allow us to exist in a functionally timeless state with God while still retaining our perception of time as finite beings and keep the energy system of the New Jerusalem stable.
So the answer to this post is yes, it could exist, it just doesn't exist yet. I think.
Alright, folks, that's my best English-major-who-grew-up-when-Evolution-was-a-fad answer to those questions. I expect the real scientists to do better. Go for it.
