The Cogito was taken to task by many philosophers but the most thorough shellacking came from Friedrich Nietzsche in Beyond Good & Evil. There are intrinsic assumptions which were exposed quite fantastically in that work. The short version is that the Cogito fails fantastically to establish existence of self.

Your question seems less about existence and more about the resolution of the Platonism / Nominalism debate.

https://philpapers.org/surveys/results.pl

Whether or not abstract things exist is at the epicenter of that debate. Do numbers exist? Do shapes exist? Do bizarre creatures concocted by one's imagination exist?

I find that religious folks tend to side with Platonism; that numbers and imagined things exist. 

Personally, that's ridiculous. Of course numbers do not exist, shapes do not exist. These are merely abstract constructs of one's mind. No more real than the boogeyman who stalks the dreams of children. It's electrical-chemical phenomenon of our brains. To say that something exists is to say that it is real, or to say that it is part of reality but our imaginations represent the imaginary or non-real. The position seems entirely bankrupt to me.

I think the issue Platonists have is thinking that properties are the same as objects. Objects have properties. An objects has properties such as shape, size, color, texture, yet Platonists think shapes are objects. That is their folly. An object can be [insert shape] but an object cannot be an [insert shape] because shapes don't exist. Objects which are circular exist, but circles do not exist. You cannot walk into the woods and pluck a circle from the weeds.

There is ample philosophy about this already and I think I've said enough to clarify my position. This forum is probably the worst place for this type of discussion. I attempted to describe the difference between intrinsic and extrinsic to somebody over several posts on another thread and the user failed miserably at comprehending this elementary distinction. I recommend reading actual philosophy (which I admit can be mentally exhausting) if you wish to further your own understanding of this debate.

That's complicated, but no, it's not literally a "ball"

The universe is expanding, which makes no sense. But (to keep it in line with my last analogy), think of it like a balloon. A balloon that never explodes (as far as we know). 

One thing that I'm curious about is what happens if you manage to reach the edge of the universe and keep going forward. It's physically impossible, since you would need to travel faster than the speed of light, but what if? According to Stephen Hawking (or what I understood from him, at least), you would just keep going around. Like the Earth, it doesn't have edges, so if you keep moving forward you would just keep going around the Earth.

Another problem is dimensions. Where is up or down in space? What is forward? The universe doesn't have a center (because every point works like a center. As in, if we say the Earth is the center, then the expansion started from the Earth. But some science stuff (radiation or something) proved that everywhere in the universe can be "the center")

But anyway, I'm looking at this in the scientific way! We exist for no reason and all at the same time. If we want to give ourselves significance, we will, but ultimately we don't have to. The universe will keep expanding, stars will die, civilizations will rise, regardless there being a reason.

... Why do you assume that the universe has an edge, or that it is finite, in any way?

Why do you think that  we need a center, up, down or forward?

Just imagine an infinite plane of little balls that repel each other.

I'm curious. If you travel faster than light, and reach a point in the universe where you can see the expansion happening, what would it look like?

The universe is finite. As finite as something infinitely expanding can be maybe the expansion will slow down, and contract in billions of years, but that's unlikely.

Yes we are.

No we're not...

Death is eternal peace, no worries, no troubles.