To get attributes to come over with the shapes in a copy and paste, there must be an exact match of the field names in the copied from and pasted into layers. Different field types or lengths can also be a problem. If you don't have a good match you will have to add fields with those exact names to the layer to be copied, use the calculator to fill them from your existing fields, and then copy and paste. Alternately, if you don't mind creating a new layer with the fields of both layers in it you can use the merge tool on selected records.