How to list all irreducible polynomials in a field?

Question

I am currently trying to refresh my memory on some basic primary polynomials, apologies if my terminologies aren't correct:

For example, I have a field $\Bbb F_{2^3}$ and generates a list of irreducible polynomails such as below $$ 1 $$ $$ x $$ $$ x^2 $$ $$ \vdots $$ $$ x^3+x^2+x+1 \ . $$

Using the example above, I have three questions:

1. Sometimes there are irreducible polynomials with cofficients such as $x^2+2$ or $2x^2+1$ , how do I go about generating the coefficients, the only way I can generate it is without coefficients

2. How do I convert something like $x^4$ into the field above?

3. Sometimes a field can be generated by the polynomial such as $1+x+x^2$. how are the irreducible polynomials generated if the field is generated by another polynomial? this question doesn't make sense

Thanks!