For the header, the easiest way is to just enter something in the custom header image blank. Alternative is to upload a transparent gif image to the images directory and enter the name of it with extension in the custom header image blank.
To remove the page title on the static home page, you would need to edit the page.php file something like this:
<?php if(!is_front_page()) { ?>
<h2 class="pagetitle">
<?php the_title(); ?>
</h2>
<? } ?>
I didn't test that code, but it should work.
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