First, it can be obtained that $a_0=1^7 = 1$.

Letting $x=1$ gives $$(1-2)^7 = a_0 + a_1+\cdots + a_7 \implies a_0+a_1+\cdots +a_7=-1$$

This means that

$$S_1 = a_1 + a_2 + \cdots + a_7 = -1-a_0 =\boxed{-2}$$

Letting $x=-1$ gives $$(1+2)^7 = a_0-a_1 + \cdots -a_7 \implies a_0 - a_1+\cdots - a_7 = 3^7$$

Subtracting this relation from the first one yields $$S_2=a_1 + a_3 +a_5 + a_7=\frac{1}{2}\times(-1-3^7)=\boxed{-1094}$$

Similarly, adding the two relations yields $$S_3=a_0 + a_2 + a_4 + a_6 = \frac{1}{2}\times(-1+3^7) = \boxed{1093}$$

Meanwhile, expanding the original expression will reveal that $a_0$, $a_2$, $a_4$, $a_6 > 0$ and $a_1$, $a_3$, $a_5$, $a_7 < 0$. Therefore, $$S_4 = \mid a_0\mid + \mid a_1\mid +\cdots + \mid a_7\mid = 1093 + 1094 = \boxed{2187}$$