The $\Sigma_n$ reflecting cardinals, of course, have
precisely the property that you mention, by definition. A
cardinal $\kappa$ is $\Sigma_n$-reflecting if
$V_\kappa\prec_{\Sigma_n} V$. One sometimes sees this term
defined in such a way to insist also that $\kappa$ is
inaccessible, but to my way of thinking, these should
simply be called the inaccessible $\Sigma_n$-reflecting
cardinals, as the concept is coherent and useful without
the extra inaccessibility hypothesis.

One can prove in ZFC, for any particular $n$, that there is
a closed unbounded class of $\Sigma_n$-reflecting
cardinals, and this is essentially the content of the
Reflection Theorem. If one adds the hypothesis that
$\kappa$ is inaccessible, then for $n\geq 2$ it follows
that there is a proper class of inaccessible cardinals and
more, but the strength is bounded below a Mahlo cardinal
(that is, very low in the large cardinal hierarchy),
because if $\delta$ is Mahlo, then $V_\delta$ has a club in
$\delta$ of $\kappa$ with $V_\kappa\prec V_\delta$, and
this club will then contain many inaccessible cardinals.

One may easily combine the reflecting idea with other large
cardinal concepts to obtain new stronger large cardinal
concepts that also exhibit your desired property. For
example, rather than considering an inaccessible
$\Sigma_n$-reflecting cardinal, one may consider a
measurable $\Sigma_n$ reflecting cardinal, or a
supercompact $\Sigma_n$-reflecting cardinal, and so on.
These large cardinal concepts will again exhibit the
property you request, but be much stronger in large
cardinal consistency strength.

Perhaps the right perspective is that the
$\Sigma_n$-reflecting idea is a way of adding "epsilon" to
a large cardinal concept. The $\Sigma_5$-reflecting
measurable cardinal hypothesis, for example, is definitely
stronger than mere measurability, but it is not as strong
as a cardinal $\delta$ that is stationary for
measurability---meaning that $\delta$ is regular and the measurable cardinals below $\delta$ are a stationary subset of $\delta$---since if the set of measurable cardinals
$\kappa$ below $\delta$ is stationary in $\delta$, then
since there is a club subset of $\delta$ of $\alpha$ with
$V_\alpha\prec V_\delta$, there will be many measurable
$\kappa$ that are fully reflecting in $V_\delta$.

In particular, if $\kappa$ has nontrivial Mitchell rank,
then the set of measurable cardinals below $\kappa$ is
stationary, and this hypothesis is satisfied. So for
example, if $\kappa$ is only $(\kappa+2)$-strong (also
known as the $P^2(\kappa)$-hypermeasurable cardinals), then
this is plenty strong enough.

The concept of a large cardinal having property $X$ and
also being $\Sigma_n$-reflecting is weaker in consistency
strength than a cardinal such that the $X$-cardinals below
are stationary, which is another common way of adding
"epsilon", in other words, of making a mild increase in
consistency strength over a given large cardinal concept.

For any fixed large cardinal concept $X$, the notion $\Phi_n(\kappa)=\kappa$ has property $X$ and is $\Sigma_n$-reflecting has the properties that you request in the second part of your question, while also being stronger than $X$. In most cases, however, this property is at the same time weaker than the typical next cardinal $Y$ above $X$, since as I mentioned, insisting on this level of reflection on top of $X$ is only adding a little, and is usually swamped by the next larger large cardinal concept. Thus, the following sequence provides a way of strengthening (in consistency strength) any given large cardinal concept:

$\kappa$ has large cardinal property $X$. (e.g. inaccessibility, measurability, supercompactness, etc.)

There are a proper class of cardinals with property $X$.

There is a $\Sigma_2$-reflecting cardinal with property $X$.

There is a $\Sigma_3$-reflecting cardinal with property $X$, or $\Sigma_n$-reflecting, etc.

There is a stationary class of cardinals with property $X$.

There is a regular cardinal $\delta$ with the $X$-cardinals below $\delta$ being stationary.

Each step increases strictly in consistency strength (for the reflecting case, you have to get above the complexity of the $X$ notion to get strict increases). So this shows that asking for reflection along with a given large cardinal notion is something in between asking for a proper class of those cardinals and asking for a stationary proper class of those cardinals. This is the same as the move from inaccessible cardinals to Mahlo cardinals, and the move from Mahlo to $1$-Mahlo, or hyper-Mahlo, and so on. But ultimately, that move is subsumed by the moves to higher levels of the large cardinal hierarchy.

It is traditional in set theory to consider individual large cardinal properties, rather than hypotheses implying a proper class or a stationary class of cardinals with the property, but these extra hypotheses are merely strenthenings of a given notion that typically have strength less than the next higher large cardinal notion.