Suppose I have the recurrence relation
$a_n = - a_{n-1} + a_{n-2} + 2^n + n$
Is it alright to solve for the homogeneous 'general' solution and then split the solving of the particular solution into two cases (one for $2^n$ and one for $n$) then combine them for the final result?
Indeed, that's just fine. Let's suppose that you've found a particular solution $b_n$ for $$b_n=-b_{n-1}+b_{n-2}+2^n$$ and a particular solution $c_n$ for $$c_n=-c_{n-1}+c_{n-2}+n.$$ Putting $a_n=b_n+c_n$ for all $n$, we have $$\begin{align}a_n &= b_n+c_n\\ &= (-b_{n-1}+b_{n-2}+2^n)+(-c_{n-1}+c_{n-2}+n)\\ &= -(b_{n-1}+c_{n-1})+(b_{n-2}+c_{n-2})+2^n+n\\ &= -a_{n-1}+a_{n-2}+2^n+n,\end{align}$$ as desired. All you have to do is add the general homogeneous solution to the particular solution $a_n$ and you're done.