Using a large resistor instead of a flyback diode
You need to be aware that the professor in the video is skipping over a few things. Note that at about 22 minutes, he writes the equation for the current through the resistor $$i = I_0 e^{\frac{R}{L}t}$$ but conveniently fails to evaluate $$ I_0= iR  $$ In other words, for his proposed 10,000 ohm resistor, a 1 amp current will provide a 10 kV voltage spike. 

Using a resistor instead of a diode is one of those entertaining tradeoffs. A diode will limit the switch voltage to only a bit over the source voltage. This is extremely useful when the switch is a solid-state device such as a MOSFET which absolutely cannot handle kV spikes. On the other hand, the voltage limit restricts dI/dt, so it takes a long time for the diode to bleed off the current. A resistor will allow much higher pulse voltages, but they will typically be much shorter. So:


  1-) If a very large resistor is used, wouldn't it be a problem if a
  continuous switching is very fast comparing to LR time constant?


Yup. You don't do it with switching regulators. The professor was referring to very low switch rates, such as applying and removing power from a circuit. Very old school.


  2-) If a diode is used, would the inductor burn if the back emf
  current is too high? In almost all examples they just use a diode
  without mentioning any possibility about this.


Nope. The "back emf current" is simply the value of the current which existed when the switch was opened. The diode needs to adequately sized, though - it must be able to carry whatever current the inductor was carrying.