You can set the CSS class for search button (SearchButtonCssClass property of search box control).
If you set the class to ‘visibility:hidden’ it will be not displayed but the button will be in the rendered code so it will work.

The problem is that often the developers are using applications like MultipleIEs to have older versions of IE on the PC. They are testing it in this application and there is probably some bug in it when using IE6. When the site is visited in the default IE6 - it works fine, but it breaks in MultipleIEs e.g. on some post back action - the CSS styles are missing or corrupted. 

Yes, Kentico CMS produces valid XHTML code. Also the built-in WYSIWYG editor automatically cleans up the code so that it's XHTML compliant. You also have full control over the HTML (XHTML) code of the page layouts and document transformations, so you can always adjust the code so that it's XHTML compliant.

If your web site uses a dark body background, you may want to use a different background for the WYSIWYG editor. In such case, you need to specify the CSS styles like these:

.LTR, .RTL
{
  // set the dark background here
  background-color: black;
}

BODY
{
  // do not set any background here ...
}

The LTR/RTL style is used only for the web page background, while the BODY style is used for both the page and the WYSIWYG editor. Alternatively, you can create a separate CSS stylesheet and assign it to the editor in Site Manager -> Sites -> Site Properties -> General dialog.