You would say cousin-in-law.

What do you call a cousin by marriage?

No, you would not - the term cousin-in-law is often used to refer to either your cousins spouse, or your spouses cousin(s) (although the more correct term would be cousin by marriage).

Can you be cousins by marriage?

In the United States, second cousins are legally allowed to marry in every state. However, marriage between first cousins is legal in only about half of the American states. All in all, marrying your cousin or half-sibling will largely depend on the laws where you live and personal and/or cultural beliefs.

What is it called if you date your cousin?

A cousin marriage is a marriage where the spouses are cousins (i.e. people with common grandparents or people who share other fairly recent ancestors). The practice was common in earlier times, and continues to be common in some societies today, though in some jurisdictions such marriages are prohibited.