Does changing the order of the steps within a round affect the security of AES?
Short answer

Changing the order of the operations does not weaken the security of $\mathop{AES}$ (nor does it increases it).

Long answer

Remark: While ShiftRows and SubBytes are commutative ($f\ g : A \rightarrow A,\ f \circ g = g \circ f$). They are not commutative MixColumns. Therefore changing the order of operations will not produce the same result. But this is not relevant because we are looking from the security point of view.

The order of the operations inside the round function does not matter. This can be shown by observing the propagation of the byte dependence over 2 application of the round function. 



In this analysis I'm ignoring the key addition because this operation can be placed anywhere in between all the operations given some tweaking. Thus we can see that the diffusion of a difference is the same as the one in the original specifications. The main reason of this result is due to the strong alignment in Rijndael.

Also this is further explained in Chapter 3 and 5 of The Design of Rijndael by their implementors.