I was pretty much in your situation 3-4 months ago
Haxe is the language and compiler, openFL is a framework/library that reproduces flash API and relies on Haxe
You write code for openFL using Haxe language. What openFL provides is all the classes you know from Flash (Stage, Stage3D, MovieClip, BitmapData, Sound etc).
as3hx conversion does not assume you will use openFL or another framework : it converts as3 code to haxe code. It will keep flash-specific stuff as flash stuff : MovieClips are still flash.display.MovieClip
This is a part that is a bit blurry for me : the whole flash API is available in Haxe, if I’m not mistaken it’s what gets called “extern classes”. If I understand correctly it’s up to the framework you’re using to bind the flash classes to relevant classes when you export.
So if you use openFL, when exporting to html5 a flash.display.MovieClip class will become an openfl.display.MovieClip class. You don’t need to worry about this, it’s automatic. If you want you can go through your code and change any “flash.x.y” import with “openfl.x.y” but I’m almost sure you don’t need to do this.
Basically once you have converted your as3 code using as3hx what you need to do is decide on which IDE you want to use (I use HaxeDevelop on windows - a fork from FlashDevelop - it works pretty well), create an OpenFL project and copy your code and assets to it.
Then edit the project.xml file at the root of the project to fit your needs, using this link as reference : http://www.openfl.org/lime/docs/project-files/xml-format/
Hope this helps a bit