Assume that
$$g_{ab}=m_{ab}+h_{ab}$$
where $m_{ab}=\text{diag}(1,-1,-1,-1)$ is the Minkowski space metric in an inertial coordinate system $x^a$, and $h_{ab$} is small and slowly varying.
$1)$ I thought that $\text{diag}(1,-1,-1,-1)$ $\textbf{was}$ $g_{ab}$???
Now the convariant metric is
$$g^{ab}= m^{ab}-m^{ac}m^{bd}h_{cd}$$
$2)$ I cannot see how the RHS of this has been derived, especially with respect to the minus sign.
1) That is only when you don't have to worry about gravity. Weak gravitational fields. That is the usual case. When you only need special relativity rather than general relativity.
2) Take the inverse of both sides of the equation $g_{ab}=m_{ab}+h_{ab}$. Rename the $a$ to $e$ here so can check whether $g_{ea} g^{ab} = \delta_e^b$. That is to expand and simplify the following:
$$ (m_{ea}+h_{ea}) \times (m^{ab}-m^{ac} m^{bd} h_{cd})\\ $$