Inductors in steady state have a constant current and thus no voltage drop across them. So just before the switch is changed, Vs must be dropped by Rs.
After the switch is moved, the current through the inductor must decay through R1.
Hint for 5-7
When working on RC transient circuits (or LR for that matter), I recommend finding the Thévenin generator to which the capacitance is connected. Remember that in general the generator may change according to the switch setting.
In this circuit the Thévenin generator is 2.7 k and 8.7 V when the switch is closed.
Hint for 5-15
Two unknowns will require two equations! One comes from setting the time constant to 13 ms for the switch in the rightmost position. The other comes from arranging for the steady-state current through the inductor (switch in leftmost position) to generate 23 kV across the 1.7 spark gap when the switch is changed.
Hint for 5-21
Once again, I would recommend finding the Thévenin generator.
Hint for 5-29
charging: How long does it take the capacitor to charge from 0.5 V to 7.0 V when connected to the 10 V source as shown?
discharging: How long does it take for the capacitor to discharge from 7.0 V to 0.5 V?