mahlo and hyper-inaccessible cardinals

$\kappa$ is Mahlo is it has stationary many inaccessibles below it, or alternatively every normal (continuous in terms of limit ordinals) function has a fixed point within $\kappa$.

This is the same because one can define a CLUB set as an image of a normal function, so having stationary many inaccessibles below $\kappa$ means there's always a fixed point.

In this way, a Mahlo cardinal is indeed inaccessible, as well as hyper-inaccessible and so on.

No amount of hyperinaccessibility or hyperhyperinaccessibility and so on can be provably equivalent to Mahloness (unless those notions are inconsistent). The reason is that if $\kappa$ is Mahlo, then all its hyperinaccessibility and hyperhyperinacessibility properties and so on are expressible in the structure $\langle V_\kappa,\in\rangle$, once one knows that $\kappa$ is regular, since they have to do only with what is happening eventually below $\kappa$. But when $\kappa$ is Mahlo, then there are many inaccessible $\delta$ with $V_\delta\prec V_\kappa$, since the set of $\alpha$ with $V_\alpha\prec V_\kappa$ is club in $\kappa$ by a Lowenheim-Skolem argument. Any such $\delta$ will therefore have exactly the same hyperhyperinacessibility properties as $\kappa$, even though when $\kappa$ is the least Mahlo, then no such $\delta$ is Mahlo. So those properties do not imply Mahloness.

Note that the property of $\kappa$ being Mahlo is naturally expressed in $V_{\kappa+1}$, since it makes reference to stationarity, which requires one to consider all subsets of $\kappa$.