You’re having an issue with PIXEL ASPECT RATIO. 960×720 is a 4:3 aspect 1280×720 is a 16:9 aspect. And that’s not just the screen’s aspect, it’s also the aspect of each pixel in the media. Do some research on non-square pixels.
When you turn Aspect Ratio Correction on, it’s doing a low-intense stretch – so you will see jaggies. But rest assured, it will look fine in export. Just for previewing.
As far as the footage looking bad in FCP, unless you need to use an alpha channel, send it out Uncompressed. Check your RT settings, you may be seeing an RT render (which will look lo-fi). Is the render bar orange?
But do some research so you know the lingo and can save time later.