Least Square Error Estimation: Conditions for $ (A^TA)^{ -1} = A^{-1}(A^T)^{-1} $?
First of all, the minimum norm least square solution is $A^+b$, where $A^+$ is the pseudoinverse. Only when the left inverse $A_L^{-1}$ (or right inverse $A_R^{-1}$) exists, you have $A^+=A_L^{-1}=(A^TA)^{-1}A^T$ (or $A^+=A_R^{-1}=A^T(AA^T)^{-1}$).

Likewise, the projection matrix onto the column space is $P=AA^+$, or $P=AA_R^{-1}=AA^T(AA^T)^{-1}=I$ if $A_R^{-1}$ exists. In other words, if A has full row rank, then the projector $P$ is indeed an identity matrix. If only left inverse exists, then $P=AA_L^{-1}=A(A^TA)^{-1}A^T$ is not an identity matrix.

Finally, $(AB)^{-1}=A^{-1}B^{-1}$ is valid only when $A$ and $B$ are square matrices.