Hi Hiker,
"When I turn off a header, its row or column returns to the default formatting for the sheet, as defined in the style."
This seems to be the default behaviour if you have formatted the table using the Format Inspector's Table section, then change the number of header rows or columns using the Table menu.
To define the style, set up a dummy table with one header row, one header column and one footer row. Put some text in each cell so you can see the effect you set for the cell fill and text colour for each type of cell, using one of the tools in the color palette.
When you have the colours set as you want them, select the table (click on any cell to select the cell, then click the bullseye at te intersection of the row and column Reference tabs to select the whole table).
With the formatted table selected, click the + sign in the top section of the (Table) Format Inspector to add that colour scheme as a Table Style.

Regards,
Barry