That's... wrong, in almost every respect.
(a) The volume is definite. It isn't "always getting bigger" - it's a specific shape, it's not changing. The only way its volume could change is if it changed shape. But it's just a static shape of infinite length. Not (as you seem to be misunderstanding) a changing shape of finite but increasing length. No, it's already infinite, it's not growing. You may be getting confused by the fact that, to calculate the volume and surface area of the infinite shape, we look at the finite-but-growing shapes, and take the limit as it grows to infinity. However, even though the finite-but-growing shapes merely approach the final volume V in the limit, the infinite shape's volme equals V (and, since the shape is static, doesn't change, so it can't be said to "approach" anything).
(b) When we look at the finite-but-growing shapes (which, again, we take the limit of in order to calculate the volume/surface area of the original shape), the volume/SA added by each additional "slice" approaches 0 for both the volume and the surface area. It definitely does not approach 1 for the SA (it's approximately a cylinder, which is 2*pi*r*h, h is a constant and r goes to 0, so the whole thing goes to 0). The statements "goes to 0" and "goes to a veryvery small number" which you try to draw a distinction between are equivalent. If the limit of a sequence is an infinitesimal, then the limit is zero. The only difference is that if you claim the limit is infinitesimal then it's evidence you're confused about something. Both the surface area and volume of each slice goes to 0, it's just that the SA goes down by 1/x, and the volume goes down by 1/x2.
(c) The limit of the length of each slice is 1, but that's irrelevant, because we're not trying to find the length of the shape. We already know that's infinite.
(d) The "no matter how far down you go, there's always more to paint" line is also irrelevant - no matter how far down you go there's always more volume, too, yet the volume is still finite. So this line proves nothing.
While no one overhear you quickly tell me not cow cow.
but how about watch phone?