Hello karim, Though generally it is possible to convert any form of energy to another(including noise) but unless you have hell of lot of noise I don't think you will be able to power an LED.
Every conversion has some losses for example if you see the energy certificate at different refrigerators you can see depending on the technology energy loss is different. So from thermodynamic point of view it is possible and if you sum all noise energy generated around the world, it may be a considerable amount. the problem is how you want to harvest it. Current technologies (at least to my knowledge) won't let you to do something useful with noise energy let alone economical. About diodes, shottky diodes have very low voltage drop. Another important parameter that you should concern is amount of leakage. shottky diodes are one of the worst diodes in this regard