Lower-income Appalachians who voted for Trump did not vote their economic interests, though it's possible that many thought they were. (Wait until Trump takes away Obamacare, but even that probably won't wing them back to the Democrats too quickly).
One reason why many people who aren't rich vote Republican is because Republicans have convinced large portions of the American public that taxing the rich will hurt everyone, and that cutting taxes for the wealthy will help lower-income people too ("trickle down" theory). My state is full of people who think like that; I'm not entirely sure how Appalachians feel about taxing the wealthy these days, though.
Somewhat true.
Appalachians very much view their economic well-being as tied to coal. Trump was pro-coal. Hillary said she wanted to shut down the coal mines. That sort of thing tends to focus the issue. Couple that with issues of gun ownership and social conservatism and the choice becomes a no-brainer.
There's lots of talk about the budget cuts and how these things hurt Appalachia. Most of these "cuts" aren't cuts; they're reductions in the rate of growth. Appalachian incomes are so low and many of the folks on public assistance there are on SSI/SSD for disabilities that these cuts aren't really going to impact THEM. Especially given that KY, WV, and OH expanded Medicaid.