Origin also allows unlimited downloads, if what was meant was installs on various computers that the player owns. It's just that only one instance of the game can be running on an internet-connected computer at a time. It's disc installs that had some sort of 5-install limit on them. Whether Origin is stable enough for anyone to want to run TS3 on various computers that way might be a different question, but it is what it is.
Yes, the base game purchase on Steam is required to purchase the EP/SPs that way.
If you have an Origin install and plan to keep it that way, you can purchase the base game (again) on Steam and then the EP/SPs that you want. They throw in an Origin code for each one at no extra charge, so you can then add the EP/SP to your Origin setup if you want. But you cannot mix Origin and Steam actual installs on the same computer and you cannot go in the other direction -- Origin codes won't work on a Steam install.