The last couple of manual (not through Magento Connect) extensions I've had to install have all had issues with the database tables not being created. These are the extensions which the installation consists of uploading two folders.
The database setup file should be triggered after a page reload so that database tables are created however they aren't. In the backend the extension shows up under System > Configuration > Advanced > Advanced – however, they doesn't show up under core_resources in the database.
Is there a setting somewhere that I've somehow flicked off? Or a way I can repair or track down what is causing these sql setup files from not being triggered?
Always install extensions with CACHE enabled. Or maintenance.flag active. In case of someone is on the site while installing.
I'd recommend you to delete all extension entries in database and core_resource table extension record. Then refresh any store page to start installation again.