Looks like you got just about everything backwards!
mfry:
- I want to use a step-down voltage regulator to adjust the input voltage of the used external PSU (to get an appropriate output voltage for the circuit). However, how is the current of the external PSU effected? Does it stay the same or is it changing due to the voltage regulator?
That all depends on what you mean by "a step-down voltage regulator". If you propose to use a"linear" regulator, then the answer is - don't! It would be a total waste of time.
It has perhaps not occurred to you that in your diagram, there is no place for your external LED supply to connect. This is because the TPIC6B595N is actually a "current sink" - it can pull cathodes of LEDs down to ground, but not supply any positive voltage to them.
So what you actually need to do, is to supply your voltage to the top rail you presently label "GND" and have all the LEDs point the other way.
mfry:
2. Because I only plan to use 3 x TPIC6B595N shift registers, a PSU which delivers (3.3V @) 450 mA would be sufficient, right?
No. Too close to the working voltage of white LEDs, might be OK for red, but I am betting you do not want to use red LEDs.
mfry:
But what do I have to do when an external power supply supports a higher current (for example 3.3V @ 1A)?
That would be very appropriate. It is reasonable design practice to use a power supply which is capable of supplying a fair bit more than you require. Two times will do quite nicely.
mfry:
3. In the schematic each LED is connected to a separate resistor. Is that the correct way to guaranty an equal illuminating power of each LED, independent of how many of them are on HIGH?
Certainly is!
mfry:
4. I am unsure which resistors I need? Is R=V/I=3.3V/0.020A= 165 ohm, right?
No, the resistors will not "see" 3.3V, they will see whatever is the difference between whatever power supply voltage you use, and the voltage drop of the LED, minus a little (albeit not much) in the TPIC6B595. The voltage drop of the LED could be 1.4V for red LEDs, about 3V for blue or white.
But what are you going to use for the actual power supply? If you use a switchmode regulated 5V supply, then for white LEDs, the voltage drop would be about 2V and a 100 ohm resistor would be appropriate. If on the other hand, you are using a 12V power supply, then 9V/20 mA is 470 ohms.
So, what power supply is it going to be?