In terms of fat loss, is it better to run early and avoid the sun, or mid day, and be exposed to the sun?

Early and avoid the sun. Sweat too much and all you'll loose is water. It's a good theory if you are going for a weigh in before an event, to sweat a lot to reduce your weight briefly. As soon as you start drinking again, it all goes back on. Plus you can seriously dehydrate yourself, which can lead to very unpleasant things happening to you.

I would advise no running at all. Activities such as Skipping, or even Fartlek Training (Switching between walking, jogging and sprinting) would be much more effective for fat loss.

First thing in the morning. it gives you metabolism a kick start and you'll be getting a little extra burn all day from it.