First, earthquakes are pretty rare down here, but they do happen once in a while. I think we've had two that I've felt at all in the ten years we've been here. Nothing has even fallen off a shelf.
As for clothes, you will probably want to invest a bit in good waterproof boots for the whole family and good outer layers that are water-proof, not just water-resistant. For just hanging around town and every day life, you will still get plenty of wear out of your cotton, but for being out in the woods or on the water, I would make sure that everyone has one or two sets of good quality long underwear and some wool socks.
WRT to furniture, that's kind of your call. I haven't really spent time in Tenakee, but I do know Juneau pretty well. (We've been here 10 and a half years.) If you have nice furniture or are particular about it, you will probably want to bring some with you. If you hit the garage sales, you can do pretty well, but it takes a lot of time and effort ('though it can be fun). I would not plan to buy new furniture here. The choices aren't all that thrilling and they tend to be over-priced.
Feel free to pm me if you need anything.