0 = ∞ + ∞ which is most certainly not true.

This is only true if both infinite are the same, but one is negative. As a general statement you're right, that this is not true.

Sure. And it also doesn't help to avoid the problems with division by zero. But lucky we're posting in the shitpost section, so we don't have to worry too much about details.

Just add ∞ to the both sides

That's why ∞ - ∞ is left undefined (same as 0/0 and ∞/∞)