(a) The greatest value is \(\boxed{5}\), and the least is \(\boxed{-1}\).

(b) The curve passes through \((0,2),(\pi,5),(2\pi,2),(3\pi,-1),(4\pi,2)\), forming one full sine cycle.

(c) The line \(y=5-2x\) intersects it \(\boxed{1}\) time. Indeed, after \(x=3\) the line is at most \(-1\), while on \([0,3]\) the difference \(2+3\sin(x/2)-(5-2x)\) is strictly increasing and changes sign.