I've created a custom block with Div and P tags in which some have id and class attributes. I've tried use FULL & Filtered html formats for the block body however the id and classes don't show up once I've added the block.
Please try change to PHP code which this option appear when you enable php filter in Modules page.
In admin/structure/types/manage/your_content_type/display (drupal 7 path) make sure field Body has format "Default" selected. If you have any other format those will remove the div ids and classes from the body.