Which Fourier Transform properties should be applied first when they are combined together? A sequence $x[n]$ has Fourier Transform $X(e^{jw})$, and the Time shfiting, Frequency shifting and Time reversal properties are shown below, respectively :
$$x[n-n_d] \Rightarrow e^{-jwn_d}X(e^{jw}) $$
$$e^{jw_On}x[n] \Rightarrow X(e^{j(w-w_O)}) $$
$$x[-n] \Rightarrow X(e^{-jw}) $$

Suppose we were given that the Fourier transform of a signal $Y(e^{jw})$ is $3e^{j4w}X(e^{-{j(w-w_O)}})$, and we want to work backwards to find the original sequence.  If I use Time reversal first then:
$$x[-n] \Rightarrow X(e^{-jw})$$
followed by Frequency shifting:
$$e^{jw_On}x[-n] \Rightarrow X(e^{-j(w-w_0)}) $$
and then finally Time shfiting and Linearity:
$$3e^{jw_O(n+4)}x[-n+4] \Rightarrow 3e^{j4w}X(e^{-j(w-w_0)}) $$

So, I have obtained $y[n] = 3e^{jw_O(n+4)}x[-n+4]$,but if I had changed the order of properties which I applied,I would have gotten a slightly different answer. Should certain properties always be applied first when dealing with such problems?
If you do things right the order doesn't matter. Note that replacing $n$ by $n+4$ in $x[-n]$ results in $x[-(n+4)]=x[-n-4]$ and not in $x[-n+4]$.