Hi David –
A simple answer is that for practical purposes, you'll want to apply the crop factor even to the numbers written on the barrel of a DX lens on a DX body. Doing so will bring you to the full frame equivalent focal length which is so common in photography nomenclature.
For example, the ideal focal length for a portrait is commonly accepted as 85-135mm, in large part because in that range, a photographer with a full frame camera must stand far enough away from the subject that facial features are not distorted. On a DX camera, the equivalent range would be attained with a focal length of ~57-90mm. (For more, see a related post, here: https://www.wyzant.com/resources/answers/21717/how_to_determine_feet_as_it_relates_to_mm_in_photography_idont_understand_and_cant_find_this_answer_anywhere_that_i_can_understand.)
Now, for techy purposes, the 18-55mm label on your lens is technically correct for your camera. Those figures refer to a distance within the camera, specifically the distance between the lens and focal plane when the lens is focused on infinity. Frankly, that is of little practical value to a photographer in the field. What we really need to know is the angle of view. The 18-55mm DX lens you mentioned provides an angle of view of 76 to 29 degrees. A similar angle of view could be produced with a theoretical 27-82.5mm FX lens on an FX body or approximated in practice with the popular 24-70mm lens.
Unfortunately, photographers don’t tend to speak of lens characteristics in terms of angle of view, though that would be more universal. Instead, photographic nomenclature is steeped in the 35mm experience. Outside of photography, people converse with degrees of view, and, for example, what is meant by a 90 degree angle of view is widely understood. However, in the world of photography, a 90 degree view would be referred to as an approximately 22mm focal length. This is because on a full frame camera focused on infinity, a 90 degree angle of view would be achieved when then lens is 22mm away from the focal plane. Seems crazy, but that is the photographic tradition.
This article has a great reference table for determining equivalent focal lengths across sensor sizes, along with the angle of view (aka field of view or FoV in the table): http://www.pentaxforums.com/articles/photo-articles/table-of-equivalent-focal-lengths.html
Finally, since your question mentioned mixing FX and DX components, I thought you might find these two videos of interest. This one, from Nikon, provides an introduction to the how and why: http://www.nikonusa.com/en/learn-and-explore/article/g588ouey/the-dx-and-fx-formats.html. The next video from Tony Northup provides additional guidance on when you would and wouldn't want to mix and match: https://youtu.be/YDbUIfB5YUc.
Hope that helps!