Hello Jennifer,
Here's my answer. You can definitely add the BillEmail field to the Invoice screen by first copying the Invoice screen and then click on edit screen, then drag the BillEmail field from the Add field option from the left pane to the screen. Once the field is added to the screen, you have to assign actions to the butCalculateAddresses button under the Add/Edit Invoice section on the screen to populate the email address into the BillEmail field. (By default the BillEmail field is not available on the Invoice screen and that is why there are no action assigned regarding that field on the screen. So once you add that field to the screen you have to add the action also to populate value in that field.)
Once you click on edit on butCalculateAddresses field, add Assign value to action result and set the action as described in the screenshot below on order no. 14
Once you assign that action, add one more action Enter Value into Field On screen to populate the email address in to the BillEmail field on order no 49.
Click on finish and Publish the screen, and test it on your Invoice screen it should populate the email address on the invoice screen. I have tested it on my account and it is working fine.
Also, you mentioned about Class Field, Yes it is available in method, click on the edit for InvoiceLineItem Grid -> go to Step 2 of 9 : Grid Columns -> scroll down to Class column and click on Adavanced -> Uncheck the checkbox Hidden on Screen first Loads -> Save & Close , Finish and Publish the screen and check your Invoice line item grid and you should have a class cloumn there.
I hope that helps!
-Inder