The picture showing your string alignment at the heel indicates that the neck needs to be realigned. You have to understand that moving the string over at the nut doesn't result in much change at the heel. Try it for yourself - loosen your E string and pop it out of the nut, towards the A string - you'll see it's still not right at the heel.
It's easy to do, as has been mentioned previously - loosen the strings a bit, loosen the neck screws a bit, push at the headstock sideways, towards the G string, a bit, until the strings are centered at the heel. Retighten the neck screws and tune. Very easy, nothing to fear.
Also keep in mind that the strings can be a lot closer to the edge down near the nut because it's a lot harder to bend in the first 4-5 frets. Up towards the middle and heel of the neck it gets a lot easier to bend, and so a lot easier to push/pull a string off the edge. I'm not saying your nut doesn't need work, it does, but I think it's a lot less of a problem than your neck alignment.