For the first video, please keep in mind that a German guy is explaining with his own quirks in his writing. For example in Germany, to represent the decimal point people use , instead of . Also in many places he writes Vr which looks like Ur and I as J.
First he shows that from the datasheet of the LED, we can see that the nominal forward voltage of the LED is 3.2V and with a current of 20 mA. (at 0.51s)
Next since he is using a 9V battery, its supply is split across the resistor and the LED. Since we know the forward voltage of the LED is 3.2V, the voltage across the resistor (which he writes as Ur) is 9-3.2 which is 5.8V. (1.38s, 1.53s)
Then in this case since the current across the resistor is the same as the LED, which is 20 mA, we can calculate the resistance by Ohm's law. R=V/I. So we get about 290 Ohm.
This is what Nardini also explained. Since the circuit is in only one steady state, I don't understand what you meant by
I'll reply to the question regarding the heat sink soon.