how to determine resistor value calculation for common anode 7-segment display multiplexed
I think you're looking for something like this


(Source)

where the individual LED current is controlled by a resistor Rs on the cathode side of each LED. (This prevents variations in LED Vf from making some elements brighter than others!)

1) No, the resistor on the cathode is used to limit current

2) You can pulse current up to the maximum pulsed current rating, which is different from the maximum steady-state current. You still must make sure that the average power dissipated is less than the maximum allowable.

For example, in this Lite-On datasheet, it gives





100mA as the peak forward current. So the maximum Rs value will be 5V-2V / 100mA = 30 Ohms (discounting Vce(sat) of T).

At that value, the dissipation in the LED will be, at most, 2V * 100mA = 200mW

This is much larger than the maximum LED dissipation of 75mW, so we should run it at a maximum duty cycle of 75/200 = 37%
(edit: the datasheet give 1/10th duty and .1ms pulse width for 100mA, so that is the limiting factor here!)

3) Determine the base resistor so that T will definitely saturate for the worst-case Beta. Assuming B=50, that would be around Ib=2mA, so Rb~2k

Note: All of this is quite back-of-the-napkin, but these values guide you through the process of getting close to the answer. In reality, this is probably much brighter/hotter than you'd ever run one of these.