learning with errors is hard even when given some information about the secret vector In this case we are given the distribution D from which secret s is sampled. If I can get some hints to proceed with the questions, it would be of help.

Thanks,
There is on one hand a recent preprint Hardness of LWE on General Entropic Distributions by Zvika Brakerski and Nico Döttling, and on the other hand it's well known that you can assume that the secret is sampled from the error distribution (see for example the last paragraph on page 7 of Classical Hardness of Learning with Errors by Zvika Brakerski, Adeline Langlois, Chris Peikert, Oded Regev and Damien Stehle).