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An asymptote.
It is called the origin, or the point (0, 0).
There is no such thing as exactly vertical because either it is vertical or it is not. You cannot have approximately vertical - it is not vertical, then. Vertical means at 90 degrees to the horizon (or horizontal).
The x-axis in coordinate geometry is the horizontal axis and the y-axis is the vertical axis.
I think {grid or grid lines} is the term you're looking for.