Example for first question:

$\Delta u=0$, say on the unit disk, with boundary condition $u_{xx}=0$. The kernel is given by linear combinations of $1$, $x$, $y$ and $xy$, so it is four-dimensional.

Example for second question:

ODE problems like $(x^2y')'-y=0$ with Dirichlet conditions $y(1)=y(-1)=0$.

Example for first question:

$\Delta^2 u=0$, say on the unit disk, with boundary condition ${\partial\over\partial n}\Delta u=0$, $\Delta u-u=0$. The boundary conditions do not satisfy the Lopatinskii condition (note that the $u$ term is lower order). Nevertheless, it follows from the first boundary condition that $\Delta u$ is constant, and then the second boundary condition implies that $u$ is constant on the boundary. So the kernel is one-dimensional.

Example for second question:

ODE problems like $(x^2y')'-y=0$ with Dirichlet conditions $y(1)=y(-1)=0$.