Hi Larry,
"Row 1 has a bottom border I can't remove, and Column A has a right border I can't remove."
Those are the borders between Header row and Header column and the body cells of the table. As you've likely learned from setting Header rows and Header columns both to 0, those borders disappear.
But a check with a table created in Numbers 3.6.2, using a template containing one header row and no header columns, showed that even those borders can be removed. Here's my test example:

The table on the left is created from a custom template as a blank table with a header row plus 10 body rows.
'Data' in the header row was entered directly.
'Data' in the body rows was created with a formula that grabbed the column letter from the entry in row 1, and attached the row number of the cell. The cells were copied, and the formulas replaced by the results using Edit>Paste Formula Results.
The table on the right is a duplicate of the one on the left, created after the last step above.
I selected the Table (click any cell, then click the bullseye at top left).
In the Format Inspector, I chose the Cell section, Clicked the top right button for All Borders, then set the border style to "No Borders".
Deselecting the table showed the result in the example above.
For two further checks, I used the Format inspector to remove the fill from cells in the header row and remove the bold attribute to the text in those cells, then duplicated the 'borderless' example, moved the original and the first duplicate down the sheet a few rows, and added a shape to the sheet and moved it down beyond the bottom of the window to make scrolling the screen possible.
On the new duplicate, I went Table (menu) > Freeze Header Row, then scrolled the screen to move the tables upward, with the result shown below:

The copy on the left shows the table after the new formatting. Note the absence of fill, bold, and row border between row 1 (still a header row) and the rest of the table.
On the right, we see the header border reappear when the frozen header reaches the window boundary and the body rows begin to slide beneath it.
Thanks for the opportunity to explore this!
Regards,
Barry