1. In your sprite collection containing the background image, go into settings, tick "Use tk2dCamera" and commit. The sprite should now be created at native size automatically, no need to click anything.
2. In the tk2dCamera, set native resolution to 1280x720, preview resolution as you have it, but you don't need to add an explicit override like that, just create a default one and leave it as is. It will automatically scale to fit content.
In both your cases, the background images should be pixel perfect.
The advantage to using the tk2dCamera is that world points = pixels, so x=100 y=50 is 100 pixels from the bottom of the screen, and 50 pixels from the left of the screen.
tk2dCamera also does scaling and other forms of aspect ratio compensation for different resolutions and aspect ratios. The override is one example. If you ran your game on a 1280x768 resolution, it would still fit the screen. It would be letterboxed, but it would still work. You'd have to implement all of that yourself with a normal camera.