Your first cousin to your children will be a first cousin once removed. Your children and your cousin's children will be second cousins to each other.
It's quite simple really. If it drops a generation on one side, then it's the same relationship but removed. If it drops a generation on both sides then it goes to second cousins: subsequent generation drops on both sides, then third cousins.
That answer was just in relation to your first question. My cousin and my husband refer to each other as 'cousin-in-law' but I don't know how correct that is.
Work out who the shared ancestor is. A shared grandparent means first cousin, a shared great grandparent means second cousin and so on. The number of "greats" indicates the cousin number. If one party has a different number of "greats" in the shared ancester then the cousinship is removed by the difference.