Lavender is caused by a recessive gene that dilutes solid black to pale gray. Solid (extended) black is dominant over patterns, so what you most likely will see from this cross is black chicks who feather in mostly solid with other colors leaking through from their mothers. I am assuming that your White Silkie is recessive white as that's the more common gene for White Silkies. White Silkies are also often carrying other color genes beneath their white outer coating, so it's hard to say what you could get from her. She may even carry Lavender and make Lavender chicks instead of Black chicks with your rooster.
As for other traits, I would expect their offspring to have crests that are smaller than their mothers', combs that are somewhere between the walnut comb of a Silkie and a rose comb, pale skin like their father, and extra toes on most of their feet. The gene that causes extra toes is weird and sometimes chicks who are a cross between extra toe and not extra toe chickens will have an extra toe on one foot and not on the other. Also, the chicks from the White Silkie will either all be bearded (if she has two copies of the gene) or half will be bearded (if she only has one copy of the gene). Since the other Silkie is non-bearded and the gene for bearding is dominant, none of her offspring will have beards when bred to a Cochin. Editing, one last important detail that somehow slipped my mind, silkied feathering is recessive, so none of the offspring of a smooth-feathered Cochin crossed to a Silkie will be silkied, but all will carry the gene.
