I'm not sure if "combine" is the right word here. If you integrate PHP, ActionScript and CodeIgniter, you'll actually be writing PHP within the CodeIgniter framework and then embedding Flash .swf files into your views.
If you plan on using this combination, what you'll need to do is map out your site and determine which parts are going to be done in PHP and which parts need to be done in Flash (or ActionScript).
My recommendation: Only use Flash where unescapably necessary. That is, use CodeIgniter for your website pages and to integrate with your data sources (via PHP) and create REST endpoints (via CodeIgniter) for data that needs to be accessed via your Flash implementations. Then, create your Flash (or Flex) application that needs to handle the audio recording and embed that into one of your CodeIgniter views.
Re:
I'll use ActionScript because I see that is most used and less complex (than FLEX).
IMO, Flex (and MXML) is essentially just a markup language for ActionScript. It actually gets compiled into ActionScript when you build your .swf file. So, choosing ActionScript vs. Flex is just a personal preference. If you're more familiar with the Flash interface than the Flex Builder interface, then go ahead with ActionScript exclusively. However, for what you say that you want to build (an audio recorder), there are already numerous examples of those available in Flex.
As far as an IDE discussion, for PHP - whatever you're familiar with. If you're looking for a full-featured IDE, I'd recommend something like PHPStorm. You can also get by with any text editor.
For Flex or ActionScript, you'd almost have to go with Flex Builder or Flash.
I'll use ActionScript because I see that is most used and less complexLess complex than what? php? javascript? most used where? not in web application, at least not nowadays...How did you form the idea to mix server-side php and Flash-integrated AS? Did you see some webapp mixing those two, by chance? Or do you just mean you want to embed some external swf in your html files? Because that's quite different...