You can't do it with just a switch and a battery. You need a circuit to sequence the LEDs. A counter with a clock or a microcontroller. Name your poison if you want some suggestions.
Don't think so. Looks plausible on paper, but third-grade math skills have a different opinion ...
Each transistor's base current is mostly the emitter current of the transistor(s) to the left. When all stages are on, the currents of 4 LED strings are going into one base.
To simplify the math, set
Vcesat = 0.0 V
Vbe = 0.6 V
LED Vf = 2.0 V
i(10K resistor) = 0 mA - the base current from the resistor is a very small fraction of the base current from the upstage emitter.
The base current into the lower right transistor is over 65 mA. Even if that approximation is off by 10-20% (and it is not), that is a very high base current given that the transistor's *collector* is rated for only 100 mA.
NOT the best.
ak
ps. Discussing this circuit would be easier if the schematic had reference designators.
pps. LTS has the Q5 base current at 44 mA, but their opinion of the transistor Vbe is very high: 0.71 V @ 0.6 mA; 0.87 V @ 44 mA. Still, that's a lotta base current.
ppps. The LED brightnesses would be better balanced if the upper right resistor were 470 ohms.