6v6gt:
OK. Thanks. Multiplexing at a granularity of segment level is certainly another method of driving a 7 segment display, particularly for reducing the number of current limiting resistors to one per digit. Interesting was that you had chosen to illuminate the same segment on multiple digits simultaneously if it was appropriate. For example, with the display of 42, as in the video, segment G (the middle segment) was lit for both digits at the same time. I'd have thought (although it didn't show in the video) that because the middle bar of the 2 was illuminated twice as often as the top bar, there would be as noticeable difference in the light intensity. Maybe it is because the camera you used did some "smoothing". If you were to display 6668 I guess the top right bar of the 8 would appear noticeably weaker that the remaining bars since it is illuminated one eighth of the time of the others.
I have only heard of it being done, but another method still using only one current limiting resistor per digit, would be to multiplex at the digit granularity level, but to use pulse width modulation to even out the luminous intensity to prevent say a 1 appearing brighter than an 8.
As for the integrated circuits to minimise the pin usage of the MPU, one other to consider is the MAX7219. There, you need only one resistor per display group and it handles the multiplexing itself.
The G segments are actually the same brightness, and the same brightness as the other segments. The current though each G segment is the same, as determined by the resistors, and that will be the case up to the point that the number of digits being driven (i.e.- the number of G segments) is greater than the port pin can source. But with these high-effiency displays, that point probably wouldn't occur until you get more than four digits. As I said in the video, I can produce "88" with this system using a total of 4ma, or 2ma per common cathode. An Arduino segment port can probably source 20ma before the voltage starts to drop.
Also, every segment is turned on for the same period of time. Remember that the segment lines cycle through one at a time, for exactly the same amount of time. So each segment lights up one interrupt out of seven. there is no difference in brightness because each segment gets the same current as determined by the resistors, for the same amount of time. The camera isn't smoothing anything. They really are equally bright. So your expectation about 6668 is not correct. Each segment is lit for the same amount of time, and they are all equally bright.
I agree that in theory you could eliminate the segment line resistors by using PWM to achieve consistent brightness, but I think you would still need the transistors, and resistors on their collector lines, unless the port pins can sink 8 segments of current at the same time. And the MAX 7219 is certainly another option, but it is a 24-pin part which I believe needs five processors pins to drive. For four digits, using the 74HC4017 with this system would use the same number of pins with a smaller, cheaper part. But it would have to be refreshed far more often than the 7219.