The matrix of generators is written as, $\begin{pmatrix} M^{00} & M^{01} & M^{02} & M^{03}\\ M^{10} & M^{11} & M^{12} & M^{13}\\ M^{20} & M^{21} & M^{22} & M^{23}\\ M^{30} & M^{31} & M^{32} & M^{33}\\ \end{pmatrix}= \begin{pmatrix} 0 & K^1 & K^2 & K^3\\ -K^1 & 0 & J^3 & -J^2\\ -K^2 & -J^3 & 0 & J^1\\ -K^3 & J^2 & -J^1 & 0\\ \end{pmatrix} $
where $J^i$ are the rotation generators and $K^i$ are the boost generators.
1)
I'm struggling top see how this can be compactly written as, $(M^{\mu\nu})^\alpha{}_\beta=i (g^{\mu\alpha}g^{\nu}{}_\beta-g^{\nu\alpha}g^\mu{}_\beta)$.
For instance, I am trying to write out the $\mu=2$, $\nu=0$ case of the matrix,
$\begin{eqnarray*} (M^{20})^\alpha{}_\beta &=& i (g^{2\alpha}g^{0}{}_\beta-g^{0\alpha}g^2{}_\beta) \\ &=& i(g^{22}g^{0}{}_0-g^{00}g^2{}_2) \\ &=& \begin{pmatrix} i & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & i & 0\\ 0 & 0 & 0 & 0\\ \end{pmatrix} \end{eqnarray*}$
but this cannot be right, it does not look like any of the Lorentz algebra elements.
Bonus question:
2) I am trying to see one obtains the operator representation of the $M^{\mu\nu}$ generators, $M^{\mu\nu}=i(x^\mu\partial^\nu - x^\nu \partial^\mu)$.
Thoughts?
I 'll just illustrate using your conventions, which I strongly suspect are in "east coast signature" (-,+,+,+), spacelike (atrocious! distinctly disadvantaged for momentum space in particle physics...), which WP follows, arguably consistently.
$$ (M^{20})^\alpha{}_\beta =-(M^{02})^\alpha{}_\beta = -(K^{2})^\alpha{}_\beta= \\ =-i (g^{2\alpha}g^{0}{}_\beta-g^{0\alpha}g^2{}_\beta) \\ = -i\begin{pmatrix} 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0\\ 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ \end{pmatrix} ~. $$
I suspect there are - signs flying around, on account of signatures changing coast during transcription. The crucial thing for you to consider is that $g^{0\alpha}=-1$ giving a net -sign in the 0th row, 2nd column entry, so a net symmetric matrix, as, of course, you should be having for boosts.
My overall correction of your formula to $$ (M^{\mu\nu})^\alpha{}_\beta=-i (g^{\mu\alpha}g^{\nu}{}_\beta-g^{\nu\alpha}g^\mu{}_\beta) $$
also agrees with $M^{\mu\nu}=i(x^\mu\partial^\nu - x^\nu \partial^\mu)$, so that $$ M^{20 }=i(x^2\partial^0 - x^0 \partial^2)= -i(x^2\partial_0 + x^0 \partial_2), $$ which sends the vector $(x^0,x^1,x^2,x^3 )^T$ to $-i(x^2,0,x^0,0)^T$, just like the above matrix does.