Magento 2 - What does the AbstractHelper do, and is it required?

Your custom class don't need to extends the Abstract helper core class.

However, the Abstract helper provides helpful methods:

• Module statuses manager - Magento\Framework\Module\Manager
• Logger - Magento\Framework\Logger\Monolog
• Http request info - Magento\Framework\App\Request\Http
• Url - Magento\Framework\Url.
• Library for working with HTTP headers - Magento\Framework\HTTP\Header
• Event manager - \Magento\Framework\Event\Manager.
• Library for working with client ip address -\Magento\Framework\HTTP\PhpEnvironment\RemoteAddress.
• Cache configuration model. Provides cache configuration data to the application - Magento\Framework\Cache\Config.
• Base64 encoder and decoder for URLs - Magento\Framework\Url\Encoder and Magento\Framework\Url\Decoder.
• Application configuration object. Used to access configuration when application is initialized and installed - Magento\Framework\App\Config

If you want to use some helpful methods, your custom class should extends from the Abstract helper.