I'd like to pick up the point you ask about in your edit.
Suppose we have a homogeneous cube and we want to construct some axes so we can locate points in the cube. The obvious choice is to put the origin at the centre and use axes that are like this:
Suppose we have two points $A$ and $B$ then we can locate them by their position vectors $\mathbf a$ and $\mathbf b$ where $\mathbf a$ would be something like $\mathbf a = (a_x, a_y, a_z)$ and likewise for $\mathbf b$. Then the interval between the events would just be $\mathbf a - \mathbf b$.
If we chose a different origin then the position vectors $\mathbf a$ and $\mathbf b$ would change but their difference $\mathbf a - \mathbf b$ would not, so the interval between the points does not depend on where we place our origin.
Now suppose we have an infinite universe instead of a finite cube. An infinite universe doesn't have a centre so there isn't an obvious choice of where to put our origin. But as explained above it doesn't matter where we put the origin because the interval between the points is independent of the choice of origin. So we can put the origin anywhere we find convenient.
So far my axes have been spatial, but in relativity time is just another axis and the universe is a four dimensional manifold. That is, we choose four axes $t, x, y, z$ and identify points by their position vector $\mathbf a = (a_t, a_x, a_y, a_z)$. So the interval between our two points will be:
$$ \mathbf a - \mathbf b = (a_t - b_t, a_x - b_x, a_y - b_y, a_z - b_z) $$
And as before this interval vector does not depend on our choice of where to put the origin. We can move the origin in space and in time and the interval stays the same. So when you ask:
suppose I have two events A and B, how can I say A occur before B
you just look at the time component of the interval, $a_t - b_t$. If this is negative then $A$ happened before $B$ and if it's positive then $A$ happened after $B$. And this result doesn't depend on what point in space or time we put the origin.
I'll mention one last point just for interest. In the above I have assumed we stay in the same inertial frame, and in that case what I've said is true that the time component of our interval is always the same. And likewise the $x$, $y$ and $z$ components of the interval don't change when we move our origin. However if we transform to a different inertial frame, i.e. do a Lorentz transformation, this causes our interval vector to rotate in the new axes. It's still the same vector, so it's length (technically its norm) is not changed by the transformation, but the individual components $(t,x,y,z)$ will be different in the new coordinates. This rotation happens in the time direction as well, so the time component $a_t-b_t$ will change. If the two points $A$ and $B$ are spacelike separated then the time component can even change sign. That is for spacelike separated points a Lorentz transformation change the time order of the events.