What they said but I’ll point out something. Since Lavender dilutes black, for the chicken to be lavender you have to have a basic black chicken to start with. Since Lavender is recessive and you are not using lavender hens you should get black the first generation with those hens.
Your lavender rooster over the barred rock hens will give you black sex links. The chicks should all be basically black but the ones with a spot on the head will be male, the ones without the spot will be female. You should be able to see the spot quite well.
Your lavender rooster over the buffs will not give you sex links. The black chicks without the spots on the head could be either male or female.
In theory all the chicks should be black, but buff can have some strange effects on black. When they feather out don’t be surprised to see orange or even yellow. That orange or yellow could be individual feathers or could come in in blotches. Some of those chickens can be quite pretty but they will not conform to an SOP.