The diagonal from $(0.1, 0.7)$ to $(-0.1, -0.7)$ has length $\sqrt{0.2^2 + 1.4^2} = \sqrt2,$ so $S$ is a square of area $1.$ The translate $T(v)$ contains a lattice point exactly when $v$ lies inside the copy of $S$ centered at that point.

Containing exactly two interior lattice points requires $v$ to lie in the overlap of two copies centered at adjacent lattice points. By periodicity the answer is the total such overlap area within one unit cell.

Consider copies centered at $(0,0)$ and $(1,0).$ Their overlap is a rectangle. A quarter-turn of the listed half-diagonal gives the relevant vertices $(0.7,-0.1)$ and $(0.3,0.1).$ Following the two pairs of parallel sides through their intersections gives side lengths $0.4$ and $0.2,$ so the overlap area is $0.08.$ Diagonally centered copies do not overlap, and horizontal and vertical adjacencies contribute equally. Accounting for the cell boundaries gives probability $2\cdot0.08=0.16.$

Thus, the correct answer is C.