You can remove All of the lines, using Layout designer.

Look at the "Attorney's" and the "Fixed Fee" invoice templates, to see how much this can vary.

Take a look at my attachment, which is an Estimate Template from QB, but it looks like a Letter.

"I mean the table that displays each invoice line item. There doesn't seem to be any way to get rid of the border lines on the table."

Click in the area in question, then click on the Properties button. Here are fonts, borders, etc.

"Also - can I just remove the header row completely from the table? I tried deleting the text from the header columns and unchecking the "fill color"box, thinking that would do it, but nope, it just puts an empty grey box over it in print preview (but its correct in the layout designer)."

You can remove the Text (label) and the formatting, and squeeze the "reserved" space to 1/16" by using the smallest resolution Snap To grid, and customers won't know you didn't "start at the top of line 1."

You will still see the pale Gray in layout designer = the Field Borders, not Printing borders. That is for your reference, to avoid positioning overlapping stuff, or to allow you to position them overlapping, if that is what you want. Intuit has overlapping elements in the templates, already.

In my class, I show that we should look at what our customer will see. Is there a reason you want to Hide so much meaningful info from them? Did you know there are formats for printing on Preprinted forms? There also is a gallery available.

"Again, I get its not Photoshop, but serious, this is basic customization for my invoices."

You have so much control here, I don't understand the issue. You can even bring in Adobe Pro .des files.

Please see my attachment.