For the first question, an inhomogeneous equation should have the same number of linearly independent solutions as the homogeneous equation. This is because if you have a particular solution to the inhomogeneous equation, you can add solutions to the homogeneous equation and still have solutions. Conversely, if you have two inhomogeneous solutions, their difference is homogeneous. Therefore you can map the inhomogeneous solutions to homogeneous solutions, so they are effectively in one to one correspondence. (This assumes that a solution exists, there are likely pathological examples that break this assumption, in which case there would be no solutions.)
In the PDE case, it is more complicated. It likely depends on the equation you are considering. To take an example, in two dimensions the Laplace equation is a second order partial differential equation. In this case, the Cauchy Riemann equations give solutions to the Laplace equation. As you may know, there are an infinite number of solutions to these equations, namely the complex polynomials $z^n$ and their linear combinations. There are likely examples with finite numbers of solutions. So in this case it is likely that there is no general answer to your question.