Adobe Forms - how to not print form field lines/shading

You could create a custom format that has outlines and shading when the field has no value, then no shading or outlines once the user enters text.  The custom format script would be something like this:

if(!event.value)
{
event.target.fillColor=color.yellow;
event.target.borderColor=color.black;
}
else
{
event.target.fillColor=color.transparent;
event.target.borderColor=color.transparent;
}

Change the colors to your requirements, of course.

I am not aware of a method to have a field show different properties when displayed on a screen vs when printed. You can however create fields that are only displayed and not printed and fields that are printed but not displayed. I would create two versions of each field in the same position: One to be filled out by the user that is displayed, but not printed, and then a second version that is read-only and set to print, but not display. You will find this option on the field properties dialog under "General>Form Field": 

The trick now is to create a caluclation option that will copy the value of the just changed user modifyable field to the print-only field. You do that on the "Calculate" tab.