Would it be possible to add a field at the bottom of a newly created item that allows us to select which takeoff folder the new item will appear in?

You can already do this. Before you click to create your new takeoff object, click on a folder on your list first so that it's selected, and then create the new takeoff object and it will automatically place the new takeoff object inside of it
However, if you're pulling from your templates, Jes's method does not work.
Yeah adding a field to assign it to a folder right on the new takeoff window would be useful for some - was just letting Kory know that you can currently do this in a slightly more tedious way