Custom Region doesnt render at the same position
You can override the block template use by your position (block--menu.tpl.php) directly in your theme folder. By default, the render of your menu must use block.tpl.php from the block module.

I recommend you to have a look at the Working with template suggestions page, and to enable the Theme debug mode to see witch template are used for your block menu.

After, to modify the Classes and the ID of your menu, I recommend you to work with template_preprocess_block hook to modify the block object before the render print it.

You can also have a look at the theme_menu_tree hook, to be able to modify a wrapper of the menu tree, depending on conditions (menu names).
There is also a THEMENAME_menu_tree__MENU_NAME possible method, that work only for one specific menu.