Question: Can I get this matrix

\begin{bmatrix}
1&1&0\\ 0&0&1\\ 0&0&0
\end{bmatrix}

from the identity matrix $I\_3$ like this:

\begin{bmatrix}
1&0&0\\ 0&1&0\\ 0&0&1
\end{bmatrix}

Add second row to first row. Then subtract second row from itself. Is the subtraction of a row from itself allowed?

Answer: 
No, you cannot subtract a row from itself!

Besides, your first matrix is *not* invertible, whereas the identity matrix is. 

Valid row operations on a singular matrix (non-invertible) will *never* yield a non-singular (invertible) matrix, while valid row operations on an invertible matrix will always yield another invertible matrix.
No, because row operations on a matrix $A$ preserve the solutions of $Ax=0$.

For the first matrix, the system is $x\_1+x\_2=0$, $x\_3=0$, which has general solution $(t,-t,0)$.

For the second matrix (the identity matrix), the general solution is $(0,0,0)$.