If you got Android File Transfer and the phone connected, you can open the Finder > (your username) > Music > iTunes > iTunes Music and you should find all your purchased music there, sorted by artist, then you can just drag them into the Android File Transfer Window which goes to your phone. Alternatively, right click on a purchased song in iTunes, do Show in Finder and there they are, and again you should be able to drag them across into your phone via Android File Transfer.
I tried Double Twist a few years ago, didn't have much luck with it then, with an Android phone and a Creative Labs media player. Not tried it since.
Here's the thing though Apple only really designs their stuff to work with their own products. You have a Mac and iTunes, Apple expects you to have an iPhone or iPad.
I'm using a Mac myself sometimes, but I've only ever had one iPhone, the original one.