Is there a way to get a column to span down rows? I would like to have a vertical menu on the left that eventually converts to a mobile button. When I am in desktop mode, the rows below the button element are dropped below the menu.

Here is one way.

Add a row with two columns. Put your menu in the left column. In the right column add a subgrid element. The subgrid will allow you to add rows while the left column is just for navigation.