1 - Silex, Slim, Lumem ... and even more robust frameworks like Symfony, Zend and Laravel can be used.
2 - It is much more feasible to use a framework for creating webservices than to use the "do it yourself" methodology, for some reasons:
- The framework has many ready-made features.
- The framework is being developed and improved over time and there are many people involved, both in the testing part and in the implementation of new functionalities.
- Another reason is that your framework done in a week will not guarantee the integrity of the app as there was no time to cover all test cases that a popular framework will achieve(will always have something to fix "here or there").
- It will take you longer to produce your own framework, time that could be spent on app features.
3 - The most correct way to understand how they work is to read the official documentation, because they have the correct practices and recommendations by the creators of the projects for better use, good studies.
- www.gajotres.net/best-available-php-restful-micro-frameworks/
- Laravel.com/Docs/5.1/controllers#restful-Resource-controllers
- Lumen.laravel.com/
- Silex.sensiolabs.org/
- www.slimframework.com/
– Junior
Not to mention that the link you passed is from 2006, good effort to search yours, and as mentioned in the question introduction, the standard will be REST.
– Junior