Random Permutation polynomial I need to know, please:

(1) Is there anyway to pick uniformly at random permutation polynomial in a field of prime order? 

(2) Are there many permutation polynomials in a field?

(3)  In a finite field of q elements how many bijective polynomials exist whose degree are smaller than d ?

***Indeed has the permutation polynomial used in this way in cryptography to generate uniformly random value? If yes, where?
Are there many permutation polynomials in a field?

For a field $F$ of order $q$, every function from $F$ to $F$ is expressable (uniquely) as a polynomial of order $q-1$.  $q!$ of these will represent permutation polynomials.

Is there anyway to pick uniformly at random permutation polynomial in a field of prime order?

Given a function from $f$ with $f(x_i) = y_i$, it's not difficult to write the polynomial that generates it.  It's just a sum of terms like $(x y_0/x_0)(x-x_1)(x-x_2)(x-x_3)... + (x - x_0)(xy_1/x_1)(x-x_2)(x-x_3)...$.  If you are generating the permutation on the fly, you can compose simpler permutation polynomials.  I don't know if there are more efficient methods.