OK, I'm not sure I have all the answers to your questions, but here goes.
As far as a background, are you referring to a full-sized render, or the thumbnail/icon for the pose? I use Netherworks MATWriter 2014 Panel, which has an option to save a thumbnail. Before I got that, I used to render at 100x100 and save as a PNG (transparent) and added my desired background in postwork. IOW, in Photoshop, PSP or Gimp, depending on your 2D graphic app of choice. If you have an actual scene set up, you can save the background when you Export > Image. It will probably default to PNG, but there's a little dropdown menu arrow along side of the box where you put in the file name, and you can choose JPG. That will save everything in the scene as rendered.
I've only saved a pose for Dawn once, and only for my own use, and I don't recall having the ground showing, or being "ticked". I think I only had Dawn selected. You may have to wait for someone more knowledgeable about saving poses to give you more info on that. As far as making the ground disappear, there are 2 methods. If you use the Hierarchy Window, you can just click on the eyeball to the left of Ground. If you don't use it, you can do so in the Properties Panel by unchecking the 3 "visible" options.
Now I'm a little confused as to what you mean by "display menu". Can you be a little more specific as to what that is? I'm using Poser9 and Poser11 Pro, and I think you're using PoserPro 2014, but I can't think of anything called a display menu. I need a little more info on that.