semiconductor physics - Does the Fermi level change under change of temperature, voltage or other conditions?

From a previous post with similar title, (What's the difference between Fermi Energy and Fermi Level?) I think it is safe to assume that

1. 
   

   In a block of material, Fermi energy is the level, up to which, electrons will fill all the available states, @ T = 0.

   
   



2. 
   

   In a block of material, in room temperature, electrons will be exited and recombined all the time. at the "Fermi level" there is 50% chance that an electron can be found there.

   
   

My questions are now

1. 
   

   For the same block of material, does the Fermi level change when the block is subject to temp change, external voltage, etc? I think it should.

   
   


2. 
   
   

   For the same block of material, should the Fermi energy should remain the same, i.e. is it an intrinsic property of that material, just like mass?

   
   


3. 
   

   When you write down the Fermi Function, in the exponential term $\frac{E-E_f}{KT}$, what is $E_f$? The Fermi level or the Fermi energy?

   
   

I am getting misleading information from the internet: http://hyperphysics.phy-astr.gsu.edu/hbase/solids/fermi.html