Question: If $f$ and $g$ have compact support, then does it follow that $f \* g$ has compact support?

Answer: 
Where you went wrong is in the second line of your reasoning: It should have read (based on your correct first line)
$$
t^2(4a) + t(4b) + 1(4c\color{red}{+2a}) = t^2(1) + t(0) + 1(0)
$$
From this, you can see that $c=-\frac12 a$ and $a=\frac14$ so one solution of the differential equation is 
$$y = \frac14 t^2 -\frac18 $$
The general solution is, of course,
$$
y = \frac14 t^2 -\frac18 + M\sin \left(\frac12 (t+\delta) \right)
$$
The forcing function on the right side is of degree two, and you'll need to add two extra degrees - one for each derivative in the highest occurrence - to do this right. That is, you need to assume $y = ax^4 + bx^3 + cx^2 + dx + e$ to start.