I have skimmed through the LAPACK user guide, but I could not find if LAPACK offers routines for Krylov Subspace based methods (such as CG or BiCGSTAB etc) and Newton method based nonlinear solvers. Also, are iterative methods available in LAPACK in general?
As far as I know, there are no such methods in LAPACK. Since LAPACK is the linear algebra package, no nonlinear solvers are included.
However, you can use the underlying BLAS for implementing iterative methods. For nonlinear solvers using Jacobians (e.g., Newton's method), the matrix factorizations of LAPACK may come in handy.
You may want to have a look at PETSc, though. This library implements many iterative methods (e.g., CG, GMRES, BiCG) and nonlinear solvers (e.g., Quasi-Newton, trust region Newton, Richardson / Picard iteration, Anderson mixing).