The first two parts (paragraphs) are ok, quite tedious but doable. I'm stuck at the third part, where V is oscillating.

I have just gone about this third part the "standard way". The coefficients for the complementary function can be recycled from the second part. Once I have substituted the RHS with V and dV/dt I get a function that has the form A sin wt + B cos wt, so that is the form of my particular integral. But the working thereafter is almost unbelievably long, like one page just to solve for one of the particular integral constants. And the answer is a big mess of Rs and Cs, which I am absolutely sure is wrong. Is there a neater way to approach this problem?

As for the final portion (at large times), I assume this follows from the general solution having the form q = exp + exp + A sin wt + B cos wt = Q_inf sin (wt + µ), so I don't think I have problems with the last portion.