Unless you want barring in there as well, it'd be best to just use Black Mottled and Lavender if those varieties are available to you. Barring from the Lavender Cuckoo offers you no benefits and part of your Lavender Mottleds from the second cross would also have barring as a result.
Neither gene is sexlinked, so it doesn't matter if you do Mottled male to Lavender female or Lavender male to Mottled female. Both will have the same result.
Yes, you would cross the siblings together from the first cross. Breeding them back to either parent risks losing the recessive gene that that parent does not have (so you could lose mottling if you breed them back to Lavender, or vice versa), whereas those birds are carrying the genes you want and will produce the variety you're looking for when simply crossed together. Inbreeding in poultry is actually not as bad as many make it out to be, and is a common practice in trying to solidify recessive traits into breeds:
https://www.backyardchickens.com/articles/inbreeding-not-as-bad-as-it-may-seem.74335/