Except you are using a 2d example to represent 3d space. This is a common problem with trying to portray infinity in a way thats understandable to everyone. We, as matter, exist with a physical 3 dimension. The existence of matter allows us to determine a center as long as we can observe that matter in it's entirety. Thus is true whether or not we live within an infinite or finite space.

If space is finite, that center is defined by its edges.

If it's infinite, and assuming that the only matter within that universe came from finite number of big bangs, then its center would be defined by the distribution of matter within it.

If it's infinite, and assuming an infinite number of big bangs, then a center cannot be determined as there exists no definable edge or boundary of matter.

Going back to your ball example and why it is a bad representation of your argument, you could define a central point. First problem, the surface of a ball is finite. Ignoring that let's look at problem two. If you add any matter to the surface of that sphere, you now have a definable point in space. As it is a single point, it would become the definable center of that surface. As you add more points, the center would be defined by a point in space that would see an even distribution of points on the surface.

That depends. Even infinite space can have a defined center if there is a finite material within that space for which we can observe distribution and find the center

That doesn't make any sense and I believe you may be misunderstanding what was being said. According to big bang theory, matter existed as a singularity that exploded into what we see as our universe. Assuming even distribution of matter, the origin of the big bang would be our center.

Except a ball would have a center as a ball is a 3d space. The center would be at the core of the ball. Same with our space. Assuming the big bang is true and the universe was created out of a singularity expanding outward, then the center would be the origin of the big bang itself. We are a point in space somewhere within the expanding matter of this universe. We are not at the center or the edge of this universe

So you think that we sit at the absolute epicenter of the universe even though we don't sit at the center if our own galaxy?

Not at all. We are the center of our observable universe and that's all. We are not the center of our universe

Trails won't last long at all with ice

I saw one break up in the atmosphere years ago. It entered as one chunk that split into 5 or 6 before burning out. Never read anything about it but everyone I told about it said it was likely mostly ice.