[...]So I gather that in order for an inverse transformation to have a directly inverse effect, it had best be applied right next to the original transformation, on either side like R-1 * R * S * T or S * T * R * R-1?
Yes, because for any matrix M
M * M-1 = M-1 * M = I
and
M * I = I * M = M
so you have e.g.
R-1 * R * S * T = ( R-1 * R ) * S * T = I * S * T = S * T
However, notice that R * S * T is an order one usually don't want to use, because scaling appears in the rotated (if using row vectors) space, so that the axes of scaling are not what one probably expects, or else scaling appears in the translated (if column vectors are used) space, what means that the center of scaling is not where one probably expects. See below for details.
I am just a bit unclear about this part. Is the transformation application order important for this decomposition method to work?
What I meant here is that the composed matrix is defined to be (still using row vectors)
S * R * T
what means that center and axes of scaling and center of rotation is not freely available.
You can use other compositions, too. For example in X3D a transform is defined as
C-1 * A-1 * S * A * R * C * T
where C denotes the center of scaling and rotation, and A denotes the axes (in form of a rotation) of scaling. Decomposing such a matrix means to not only look for S, R, and T, but also for C and A and hence is more complex than what I have shown.