So for regular mottled, if I got a black mottled, I couldn't breed black mottled to black mottled correct? I would need to breed it to black first?
No, you absolutely can breed two mottled birds together. What I meant originally was in the case if you only had one black mottled bird and wanted to breed more. Mottling is recessive and breeds true, so breeding together mottled chickens is the best (and easiest) way to go.
Then the mauve mottled, would I start with black mottled and breed that to blue and chocolate? Would this combination also give some chocolate mottled as well?
There are multiple ways you could go about it. The easiest would be to just get mauve mottled chickens and breed them together, but that's not the only way to do it, and there are alternatives, in case you cannot get mauve mottled.
If you bred black mottled to blue and chocolate, you'd get some blacks, some blues, some blacks carrying chocolate, and some chocolates, depending on what exactly you crossed. All chicks from the mottled parents would carry mottling, so they won't be mottled themselves, but they could pass it onto their chicks.
Black mottled x blue = 50% black, 50% blue, all carrying mottling (which parent is the roo and which is the hen doesn't matter)
Black mottled (hen) x chocolate (roo) = 50% chocolate females, 50% black carrying chocolate males, all carrying mottling
Black mottled (roo) x chocolate (hen) = 50% black females, 50% black carrying chocolate males, all carrying mottling
At this point, you wouldn't have any mauve or mottled chicks yet. You could now take the blues that carry mottling and breed them to the chocolates and chocolate carriers that carry mottling.
It would look something like this:
Blue carrying mottled (roo) x chocolate carrying mottled (hen) =
females: 6.25% black, 12.5% black carrying mottled, 6.25% black mottled, 6.25% blue, 12.5% blue carrying mottled, 6.25% blue mottled;
males: 6.25% black carrying chocolate, 12.5% black carrying chocolate and mottled, 6.25% black mottled carrying chocolate, 6.25% blue carrying chocolate, 12.5% blue carrying chocolate and mottled, 6.25% blue mottled carrying chocolate
Blue carrying mottled (hen) x black carrying chocolate and mottled (roo) = females: 3.13% black, 6.25% black carrying mottled, 3.13% black mottled, 3.13% blue, 6.25% blue carrying mottled, 3.13% blue mottled, 3.13% chocolate, 6.25% chocolate carrying mottled, 3.13% chocolate mottled, 3.13% mauve, 6.25% mauve carrying mottled, 3.13% mauve mottled;
males: 3.13% black, 6.25% black carrying mottled, 3.13% black mottled, 3.13% blue, 6.25% blue carrying mottled, 3.13% blue mottled, 3.13% black carrying chocolate, 6.25% black carrying chocolate and mottled, 3.13% black mottled carrying chocolate, 3.13% blue carrying chocolate, 6.25% blue carrying chocolate and mottled, 3.13% blue mottled carrying chocolate
I'm sure it could've been summarized in a simpler way, so sorry if it looks a bit messy.
So, yes, you could get chocolate and mauve mottled chicks if you bred a blue hen that carries mottled to a black roo that carries chocolate and mottled. But you would also get a bunch of other colors.
In breeding these, do certain colors need to be roosters or hens for certain genes to show and be passed down?
It starts to matter when chocolate gets involved, because it's sexlinked recessive. I did include both hen and roo versions in my pairing examples above, which show the different results you'll get, depending on which parent is the male and which is the female. If you have a question about a specific pairing that I did not include here, let me know.
I know this is a lot, and I hope I haven't missed anything. Hope this helps!