1. Just evaluate $\langle \lambda, \alpha \rangle$ for all roots $\alpha$. It is sufficient to test the equality for positive roots only. Also, the weight $\lambda$ is regular if and only if it has trivial stabilizer in the Weyl group. For example, if you write $\lambda$ in Bourbaki's "$\epsilon$-basis" then regularity means that no two coefficients are the same.

2. If the weight is regular, then index equals the length. For singular weights I think you need minimal length representative of cosets $W/W_S$ where $W_S$ is the subgroup corresponding to the Levi part of parabolic subalgebra that stabilizes $\lambda$.