If your location is Chapel Hill, NC (assuming your 'zone' listing is really your zip code), then your fig tree likely was frozen back and damaged by the hard winter this year. Give it some time, see how it bounces back. Later in the season, maybe early summer, prune out what wood is really dead. And as noted above some backup cuttings, so you can replant or have more trees would not be a bad idea.

It is hard to tell them apart. Because you said that the leaves looked OK earlier it is most likely mites. FMV will often show early in the season and then get better, while mites cause more symptoms as the season progresses and their population increases. They are the vector of FMV so be careful to not spread them from tree to tree on your hands and clothes. It seems like Avid is natural but not organic for some reason. Be careful to not get in your eyes or on your skin, read the label, get very good coverage- inside to outside, bottom to top.

It depends. Treated municipal water might contain chlorine (often not the kind that dissipates after the water stands for a while), and hard water can cause harmful mineral deposits, but you can avoid that by flushing the soil frequently. Still, rainwater or R/O (reverse osmosis) water is great, especially to flush accumulated mineral deposits from soil. Just be aware that if your ficus's roots can't take up water at all, either because they've died of dehydration or from suffocation because water retentive soil stayed soggy for long periods of time, even the purest water won't help. That being said, one of the easiest things to start with is flushing the soil to remove mineral deposits, and R/O or rainwater make that easier. I used R/O water from a supermarket machine to flush my plants, but when I'm not up to hauling stuff I use tap water from my faucet mounted filter. Once you've done that, learned how to tell your plans actually needs water before watering, and learned how to remove as much perched water as possible after watering, you can focus on getting your ficus into a fast draining soil that doesn't pose a danger to its roots. Another excellent thread called Dealing with water retentive soils: http://forums2.gardenweb.com/discussions/1415041/dealing-with-water-retentive-soils?n=62

Most likely: A complex chain of events that results in shedding of leaves can be set into motion when the photo load (amount of light) a plant is receiving is changed dramatically. It's especially noticeable when the photo load is diminished. Older/lower leaves are usually the first victims. Other potentialities: Over or under-watering can cause shedding of foliage, as can nutritional deficiencies of the nutrients considered mobile in the plant (mostly N, P, K, Mg). Root congestion and soil compaction are other possibilities. Al