The fact that you are dealing with compact and/or finite dimensional Lie groups is completely irrelevant. The fact that these group are Lie is also partially irrelevant (unless you care about putting connections on your bundle gerbes, in which case it becomes very relevant).
More relevant is whether the groups abelian or not.
A priori, the cocycle relation only makes sense for abelian groups.
But there is also a theory of non-abelian (bundle) gerbes, where you allow non-abelian groups. The cocycles have two kinds of data: Maps
$\alpha_{ij}:U_i\cap U_j\to \mathrm{Inn}(G)$ and maps
$g_{ijk}:U_i\cap U_j\cap U_k \to G$,
where $\mathrm{Inn}(G)$ denotes the group of inner automorphisms of $G$.
These non-abelian gerbes are classified by $H^2(-,Z(G))$, the second Cech cohomology group with coefficients in the sheaf of $Z(G)$-valued functions. [that's a non-trivial theorem]
That was the case of a trivial band.
A band is the same thing as an $\mathrm{Out}(G)$-principal bundle.
Say you are given an $\mathrm{Out}(G)$ principal bundle $P$, described by transition functions
$b_{ij}:U_i\cap U_j\to \mathrm{Out}(G)$. Then you can twist the above definition as follows:
The cocycles now consist of maps
$\alpha_{ij}:U_i\cap U_j\to \mathrm{Aut}(G)$ and maps
$g_{ijk}:U_i\cap U_j\cap U_k \to G$,
where the $\alpha_{ij}$ are lifts of the $b_{ij}$.
The gerbes with band $P$ are classified by a set that is either ♦ empty, or ♦ isomorphic to $H^2(-,Z(G)\times_{\mathrm{Out}(G)} P)$, the second Cech cohomology group with coefficients in the sheaf of sections of
$Z(G)\times_{\mathrm{Out}(G)} P$.
Whether or not that set is empty depends on the value of an obstruction class that lives in $H^3(-,Z(G)\times_{\mathrm{Out}(G)} P)$.
It's non-empty iff that obstruction vanishes.
Finally, to answer your last question.
If $G$ is a Lie group and you have a bundle gerbe with connection (trivialized over the base point), then you get a $G$-principal bundle, but only on a subspace of the based loop space $\Omega M$.
It's the subspace consisting of those loops over which the band $P$ and its connection trivialize.
Consider the special case $M=\mathbb R^2$ and $X\equiv(1,0)$.
Now, given any vector field $Y$ in terms of component functions as $Y(x_1,x_2)=(y_1(x_1,x_2),y_2(x_1,x_2))$, a simple calculation gives
$$
ad_XY
=
(\partial_1y_1,\partial_1y_2).
$$
This means that $ad_X=\partial_1\otimes I_2$.
Here $I_2:\mathbb R^2\to\mathbb R^2$ is the identity operator.
(In the Euclidean space we can differentiate somewhat carelessly and freely identify tangent spaces with their duals.)
Given any $x\in\mathbb R^2$ and $\xi\in T_xM$, the principal (and full) symbol is
$$
\sigma_{ad_X}(x,\xi)
=
i\xi_1\otimes I_2
:
\mathbb R^2\to\mathbb R^2.
$$
This is invertible if and only if $\xi_1\neq0$.
In particular, at any point $x$ there are non-zero tangent vectors $\xi$ (e.g. $(0,1)$) for which the principal symbol is not invertible (it vanishes entirely!).
Therefore the operator is not elliptic in the sense of invertible principal symbols.
This special case is actually not special.
The same argument works in any $\mathbb R^n$ or an open subset.
If $X$ is a non-vanishing vector field in an open subset of a manifold, you can set up local coordinates so that $X$ becomes a constant vector and the same argument goes through.
Best Answer
The book of Giraud is a fundamental reference on the subject, but you have to be used to the language of Grothendieck. A reference more accessible, for example for a differential geometer is the chapter 5 of the book of Brylinski which deals only with commutative gerbes.
J.L Brylinski Loop Spaces, Characteristic Classes and Geometric Quantization.