Dissect a cylinder first!

Let's assume that it is possible. Then since the surface area $X = 2\pi rh + 2\pi r^2$ , we would have

$\frac{X}{2} = 2\pi r\frac{h}{2} + 2\pi r^2$

$\frac{2\pi rh + 2\pi r^2}{2} = 2\pi r\frac{h}{2} + 2\pi r^2$

$\pi rh + \pi r^2 = \pi rh + 2\pi r^2$

$\pi r^2 = 2\pi r^2$

$r^2 = 2r^2$

$r^2 = 0$

$r = 0$

which does not make a physical cylinder. Therefore, by contradiction this not possible .

The total surface area of cylinder is given by : $S.A = \underbrace{2 \pi rh}_{\text{curved surface area}} + \underbrace{2 \pi r^2}_{\text{lateral surface area}}$

So, if height is halved then only curved surface area gets halved but still lateral surface area $(2 \pi r^2)$ will remain same as it is independent of height. So, the total surface area will not be halved if height is halved.