Probably because they don't have a reliable way of differentiating one strand of the double-helix from the other, so matching bases are considered to be a single unit so that it doesn't matter how you've got the pair rotated.

I imagine this could be solved by ensuring that every helix starts out with something like a byte order mark that would distinguish the two strands reliably.

Storage density could be further increased by a constant factor if more kinds of bases were used, or if they could use single-strand DNA/RNA instead (which would probably require some chemical means of ensuring that a free strand doesn't accidentally bind to something else).

i answered this above; basically to avoid certain problematic sequence features.

Bases can only pair with one other base. Adenine can only pair with thymine. Guanine can only pair with cytosine. Knowing one base implies the other, so there are only two possible pairs.