I am writing code for solving linear equations of the form

$$A_{n\times n}\cdot x=1_n$$

where $n$ is on the order of $10^6$ and $A$ is a symmetric matrix with approx $10^3$ nonzero entries in each row. This makes its size barely manageable, but inverting it is infeasible, and I'm not sure which decomposition suitable for solving linear equations would lead to sparse matrices.

Thus the questions:

1) Any hope that $LU$ decomposition of a symmetric sparse matrix would be sparse?

2) Is it possible to take advantage that r.h.s. is a scalar to simplify solution of the above equation?

3) What would be the best numerically stable algorithm to handle linear equations of that size?