If you are prepared to go digging around in the fortran code:
The SVD algorithm consists of a few parts:
Bidiagonalization (usually using Householder reflectors)
Use QR shifts to reduce the bidiagonal to diagonal (usually using Givens rotations)
Change the sign of any negative diagonal elements and reorder the singular values so the are decreasing.
For part 1: To my knowledge, the answer is no.
For part 2: This question had several good answers, all of which were negative (there isn't really a faster way). I don't believe there is meaningful new literature on the topic.
Using the Cholesky decomposition is the quickest way I know of checking if a symmetric matrix has negative eigenvalues. Nothing wrong with that. Plus, if it succeeds, you already have the Cholesky decomposition! Of course, if there are any negative eigenvalues, it will fail. You will not need to check the entries of $L$.
Cholesky factorization is for symmetric positive definite matrices, and it will fail if the matrix has negative eigenvalues. You should use singular value decomposition for that purpose, or maybe a QR algorithm would suffice if you just need some of the eigenvalues.
Edit 1: Of course, I should also add that, in general, L would not give you much information ...