Sigh. Take this part:
I'm sure the point has been repeatedly made in this thread that this is completely wrong.
You are, as is almost always the problem. Using your intuition about what happens at every stage after a finite number of decimal places to assert something about the infinitely long decimal expansion. Your intuition is wrong. Infinity is not 'a really big real number'. It is not a real number.
Parallel lines do not meet in the Euclidean plane. If I'm wrong (and I'm not), then feel free to write down the point of intersection: hint there is no such point as infinity on the Euclidean plane.
The place to use points at infinity is projective geometry, and there need not be just one point at infinity.