Answer
a. Z=4 and Z=20.
b. $1s^4 2s^4 2p^3$.
Work Step by Step
In this alternate universe, $m_s$ can have four different values: $-\frac{3}{2}, -\frac{1}{2}, \frac{1}{2}, \frac{3}{2}$. Each $nlm_l$ state can hold four electrons, each one having a different $m_s$ value.
Apply the exclusion principle.
a. For a filled n=1 shell, the electron configuration would be $1s^4$. Z=4.
For a filled n=2 shell, the electron configuration would be $1s^42s^42p^{12}$. Z=20.
b. Sodium has eleven electrons. The ground-state electron configuration would be $1s^4 2s^4 2p^3$.
The periodic table would be very different.
