Minimum Nonlinearity of boolean function
You can give a boolean function by it's truth table.
For example if $f(x, y, z) = x + z$ you have the following truth table:

  x   y   z   f(x, y, z) = x + z
 ----------------------
  0   0   0     0
  0   0   1     1
  0   1   0     0
  0   1   1     1
  1   0   0     1
  1   0   1     0
  1   1   0     1
  1   1   1     0


just enumerating the last columns gives you $(0, 1, 0, 1, 1, 0, 1, 0)$. This $f$ is by its definition clearly a linear function. If you want to enumerate all linear functions you can look at the Hadamard matrix and Hadamard transform.

To calculate the linearity of a boolean function $g$ as explained in this question you take the minimum Hamming distance between the output vector of $g$ and all linear functions.

So if your function would for example be $g = (1, 0, 0, 0, 1, 1, 0, 1)$ the Hamming distance to $f$ would be.

g: 1 0 0 0 1 1 0 1
f: 0 1 0 1 1 0 1 0
------------------
   1 1 0 1 0 1 1 1 =&gt; HD 6 =&gt; non-linearity at most 6


In fact if you where to try all linear functions you would find $h = (0, 0, 0, 0, 1, 1, 1, 1)$ with 

g: 1 0 0 0 1 1 0 1
h: 0 0 0 0 1 1 1 1
------------------
   1 0 0 0 0 0 1 0 =&gt; HD 2


and this is the lowest value so the non-linearity of $g$ is 2.

I hope this answers your second question.

Now we can answer your first question simply by finding examples.
Here are two 4-bit functions with different non-linearity:


[1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0]
Non-linearity here is 2. It cannot be more than 2 because the middle two bits are swapped from

[1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0] 
which is clearly linear (depends only on highest value bit). And it cannot be less than 2 because the function is non-linear and balanced.
[1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0] has non-linearity 4.


This, I hope answers your first questions negatively. Not all functions have the same non-linearity.