KCL for this Supernode
For starters, I don't understand how the numerator on these fractions
  ends up as anything but one....


I've left several comments pointing out careless errors in your equations but this deserves an answer even if the answer is elementary algebra.

Take, for example your first (corrected) equation:

$$\frac{1}{10}v_1 + \frac{1}{20}v_3 +\frac{1}{2}(v_3-v_2)+\frac{1}{2}(v_1-v_4)=0$$

Group the terms involving, for example, \$v_1\$

$$v_1(\frac{1}{10} + \frac{1}{2}) = v_1 \frac{3}{5}$$