How can I set Numbers to automatically copy a row of data from one sheet to another sheet?
I don't know if it's possible to do this with functions, but it might be. There is a function called VLOOKUP that seems like it would do what you want, but I couldn't get it to quite work in my experimenting with it—a function that exists within one cell has a hard time returning multiple values for multiple cells.

What I think does work, though perhaps not as convenient as you might want, is the filter view. On the right side of the window, if you select the view that says "Sort &amp; Filter" and the Filter tab, you can "Add a Filter" that will match a value in a specified column. Toggling that filter using the checkbox in the corner of that Filter's box will hide all of the rows that don't match that criteria, then it's a simple matter of highlighting all of the cells, copying and pasting to another sheet. 

Unfortunately, this won't continue to add rows as you add rows in the first sheet, so if that functionality is something you need, this won't answer your question. If you only need a one-time copy, though, I think that this is the easiest option (and the only one that I know of, but I'm not extremely familiar with Numbers).