Is Morrowind Stargate free?

Yes. Morrowind Stargate is released under Free fan mod (see the ModDB page for terms) and costs nothing. Neither it nor The Elder Scrolls III: Morrowind has any paid content.

What do I need to run Morrowind Stargate?

The Elder Scrolls III: Morrowind, which is itself free. Morrowind Stargate is an add-on and will not run standalone.

How do I install Morrowind Stargate?

The PlayBound Launcher installs it in one click from The Elder Scrolls III: Morrowind's mods list. You can also install it by hand — full steps are on this page.