The
TV you got is native 720p (well, 1366x768 per usual there). It's not uncommon for a TV to accept 1080p and scale it though, which is likely why you see it offer that resolution. You'll want to keep it at native.
As mentioned, you'll want to turn off overscan to avoid the picture going off screen. This looks to be something you can toggle (
page 22 of your manual), though the VGA input should have it off by default.
Some of the other picture quality issues may simply be a matter of adjusting settings. For instance, you may try turning down Sharpness as that tends to introduce haloing, especially on text.