The first question: what is the proof that LU factorization of matrix is unique? Or am I mistaken?
The second question is, how can theentries of L below the main diagonal be obtained from the matrix $A$ and $A_1$ that results from the row echelon reduction to $U$? ($A=LU$)
The factorisation is not unique. There are $n^2+n$ coefficients to estimate and only $n^2$ "equations". As such, that is why there are the two "common" methods, Doolittle and Crout see wiki page. For each of these two approaches, you can show that the resulting linear system has a unique solution.