Ugh, sounds like the patch apply from hell... I can give you a list of all the files that are patched. They should have the file date of the day you successfully applied the patch and should be listed in app/etc/applied.patches.list
A sampling of the dates on 6-8 files should tell you if there was any activity towards having the patch run.
If it dies on one file, it applies nothing at all. It's purposefully an all or nothing operation so it will not leave you in a half-dead state. Given the way this particular one works (SUPEE-6788), one of the security issues is cross-module and a partial apply will leave your site a heap of rubble.
As to application, this patch must be applied with the Magento cache fully disabled, cleared and the compiler shut off. People have had problems where they didn't do one or the other of the above and the table installer scripts fail to create the whitelist tables, even though they record the successful run of the installer scripts. Also, after you install, log into admin, log out and then back in to make sure the menu items actually load. Sounds like it's not a first time admin backend load missing menu items though.
You might prod Magentary to see if they will do the patch bundle for 1.6.2.0 which would reduce it to a simple file upload instead of a patch run.
.htaccess
.htaccess.sample
app/code/core/Mage/Admin/Model/Block.php
app/code/core/Mage/Admin/Model/Resource/Block.php
app/code/core/Mage/Admin/Model/Resource/Block/Collection.php
app/code/core/Mage/Admin/Model/Resource/Variable.php
app/code/core/Mage/Admin/Model/Resource/Variable/Collection.php
app/code/core/Mage/Admin/Model/Variable.php
app/code/core/Mage/Admin/etc/config.xml
app/code/core/Mage/Admin/sql/admin_setup/upgrade-1.6.1.0-1.6.1.1.php
app/code/core/Mage/Admin/sql/admin_setup/upgrade-1.6.1.1-1.6.1.2.php
app/code/core/Mage/Adminhtml/Block/Permissions/Block.php
app/code/core/Mage/Adminhtml/Block/Permissions/Block/Edit.php
app/code/core/Mage/Adminhtml/Block/Permissions/Block/Edit/Form.php
app/code/core/Mage/Adminhtml/Block/Permissions/Block/Grid.php
app/code/core/Mage/Adminhtml/Block/Permissions/Variable.php
app/code/core/Mage/Adminhtml/Block/Permissions/Variable/Edit.php
app/code/core/Mage/Adminhtml/Block/Permissions/Variable/Edit/Form.php
app/code/core/Mage/Adminhtml/Block/Permissions/Variable/Grid.php
app/code/core/Mage/Adminhtml/controllers/Permissions/BlockController.php
app/code/core/Mage/Adminhtml/controllers/Permissions/VariableController.php
app/code/core/Mage/Adminhtml/etc/adminhtml.xml
app/code/core/Mage/Catalog/Model/Product/Option/Type/File.php
app/code/core/Mage/Core/Controller/Request/Http.php
app/code/core/Mage/Core/Controller/Varien/Router/Admin.php
app/code/core/Mage/Core/Helper/UnserializeArray.php
app/code/core/Mage/Core/Model/Email/Template/Filter.php
app/code/core/Mage/Core/Model/Resource/Setup.php
app/code/core/Mage/Core/etc/config.xml
app/code/core/Mage/Core/etc/system.xml
app/code/core/Mage/Customer/Block/Account/Changeforgotten.php
app/code/core/Mage/Customer/Block/Account/Resetpassword.php
app/code/core/Mage/Customer/controllers/AccountController.php
app/code/core/Mage/Downloadable/Model/Product/Type.php
app/code/core/Mage/Eav/Model/Resource/Attribute/Collection.php
app/code/core/Mage/Sales/Model/Resource/Order/Item/Collection.php
app/code/core/Mage/Sales/controllers/DownloadController.php
app/code/core/Zend/Soap/Server.php
app/code/core/Zend/Xml/Exception.php
app/code/core/Zend/Xml/Security.php
app/code/core/Zend/XmlRpc/Request.php
app/code/core/Zend/XmlRpc/Response.php
app/design/adminhtml/default/default/layout/admin.xml
app/design/frontend/base/default/layout/customer.xml
app/design/frontend/base/default/template/customer/form/register.phtml
app/design/frontend/base/default/template/customer/form/resetforgottenpassword.phtml
app/design/frontend/base/default/template/page/js/cookie.phtml
app/design/frontend/base/default/template/persistent/customer/form/register.phtml
errors/processor.php
lib/Unserialize/Parser.php
lib/Unserialize/Reader/Arr.php
lib/Unserialize/Reader/ArrKey.php
lib/Unserialize/Reader/ArrValue.php
lib/Unserialize/Reader/Bool.php
lib/Unserialize/Reader/Dbl.php
lib/Unserialize/Reader/Int.php
lib/Unserialize/Reader/Str.php
lib/Varien/Data/Collection/Db.php
Been dealing with module testing on staging all day to make sure nothing's busted before I consign myself to hell.
which means the patch was not applied... last solution I'm thinking of is editing the content from the patch deleting lines in relation with the htaccess as explained here although I don't like to do that.... http://www.atwix.com/magento/security-patch-supee-6788-installation-issues/