1) As with almost everything in TPC, there is more than one way to do this but the easiest and simplest way is to right-click the object and choose Modify Objects | Toggle Space. If the object is in Survey Space, it will toggle into Paper Space and vice-versa. You can even use Edit | Select Objects to select any number of objects in any or both spaces to toggle them all at once. You don’t need to do them individually.
2) This one is a little tougher and I can’t answer it directly.
Because traverses pretty much replace the way that layers are used in a traditional CAD program, we don’t worry about what layers they are on in the Drawing View and there is no way to change it.
In most cases, this question comes up when someone needs to send a file to one of those poor people who are forced to use a traditional CAD program and they want/need certain things on certain layers. If this is the situation you are talking about, this is handled with some setup on your part and the layers are created in the exported CAD file. You will set the layer names that are needed either in the Traverse Properties dialog box on the Traverse tab or in the Traverses Manager in the Layer column. (Note that you may need to Tools | View | Format View in the Traverses Manager and add an L to the Display Sequence to display the Layer column.) You will also need to make sure that Use Traverse Layers is turned on in the export settings for the file type you are exporting and you need to make sure that you understand the wildcard options that are available for the new layer names.
There is a lot of information on this topic in the DXF / DWG Export Settings Dialog Help topic. You can access this Help topic from the Drawing View by choosing Tools | Export Drawing As, setting the Type to AutoCAD DWG, clicking the Settings button to the right of the Type (it looks like a gear) and hitting the F1 key on your keyboard.
Thanks for the great questions and let me know if you need more information or if I need to clarify anything.