Re: What. A. Shock.
"There are obvious problems with infinities"
The biggest problem with infinities is our ability to understand them. As with quantum physics, if you think you understand it, you're probably wrong. And as is so often with physics outside the familiar human scale, this tends to lead to our intuitive understanding getting things completely wrong. Without going into details on all your points, I'll just note that the Big Bang is entirely compatible with an infinite universe. That is, after all, why people are still arguing about whether the universe is infinite or not, despite everyone involved being pretty sure the Big Bang did happen.
I will correct you a bit on the horizon thing though. The horizon defined by the point where things start moving away from us faster than light is called the Hubble horizon, and it is not related to the size of the observable universe. The size of the observable universe is defined by the particle or cosmological horizon, which is simply a measure of how far light has been able to travel since the universe began. Obviously the past expansion of the universe affects this, but whether it's currently expanding or not, and whether it's accelerating or not, is completely irrelevant. It's possible for an object to be beyond the Hubble horizon, but still be inside the particle horizon.