How do you remove  the home  tab from the top  navigation menu? We
already have a logo in the top left corner that links to home, so
having it in the menu  is a waste of good real estate.

We'd also like to remove the little down arrow that appears on tabs
that have subsites, and have had no success. Is it possible to remove
that?

Take a look at the following BLOG entry from the ECM team at Microsoft.
You can remove  the home  tab by setting the ShowStartingNode attribute of
the TopNavigation bar to False and remove the arrow by setting the
MaximumDynamicDisplayLevels to 0. Of course if you do that then you'll
also disable the dropdown list of subsites. I don't think there is a
way to remove the arrow and show the dropdown without building your own
custom control.

Here's the Blog that explains all this and more:

blogs.msdn.com/.../moss-navigation-deep-dive-p
art-1.aspx

The down-pointing arrow on the top  navigation tabs indicates that this
tab includes sub-sites. You can turn off the arrow.
Navigate to the Tab
Select Site Actions
Select Site Settings
Select Navigation under Look and Feel
Unselect the Show Subsites check box
Select OK.

Interested in seeing the response to the first question. We have the
Logo, a breadcrumb, a tab  all within a square inch of screen space and
all roads lead  to the same place.

To remove  the arrow:

You should be able to do it with styles/css. This works  but not when
the topnav is selected. I dont know why.

Put the following html  last in the head tag  in your master-page:

<style type="text/css">
.ms-topnav img, .ms-topnavselected img
{
display: none;
}
</style>

Even though the selected  does not work I hope you understand the
principal of my solution. Maybe this can lead  the way though