1. Not sure what you mean by valid. "Loss harvesting" and "gain harvesting" are both valid tax planning moves. If you sold all of your FXNAX position, from your own description you would realize about $6,650 in long term capital losses, and about $350 in short term losses.
2. I think it is not a wash sale, and I think most would agree with me.
3. If you want to sell, there's no reason to. Short term losses are, in general, more valuable than long term losses, so if anything, it'd be better to sell sooner rather than later for lots where you have losses. Your short term losses will offset any short term gains first, then any long term gains second (sounds like you don't have any of these).
Short term gains are taxed at ordinary income tax rates, so they're usually best avoided, but if it's a small lot from dividend reinvestment or something like that and you value simplicity, it's probably not that big of a deal.
4. I think the default depends on your broker. Ask them.
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