Insert, update and get in the same php file?
Your concept using the HTTP verbs is pretty much the idea behind REST. You could go one step further and use the less known verbs like PUT, DELETE, UPDATE. 

Also u could use the $_SERVER['REQUEST_METHOD'] to switch between the HTTP modes you would like to process. 

I would suggest to write a "class Request" to be reused by all your CRUD pages or use / get inspired by (for example) symfony2 

https://github.com/symfony/symfony/blob/master/src/Symfony/Component/HttpFoundation/Request.php

u are using classes like CategoryMySqlDAO why do not u use a CRUD class? u could implement an abstract class CRUD with "create, update, delete" methodes and then use subclasses for each crud page. Then you would inject the DAO and stuff in the constructor and use autoload to get them instead of a smelly required statement in each file. 

also having HTML and logic in the same file never is a good idea. 

i hope i could give you some inspiration how to improve your current state