If you wind the strings properly you only need 2 winds on the peg. Much else leads to slippage. I only put 1 wind on me E&A strings and suffer no adverse effects.
Locking tuners dont require 1 complete wind making restringing very quick, stopping the string binding on itself and therefore creating better tuning stability.

The only time i've needed a tree to stop nut ring is when the nut is worn. Then you cant raise the tree much at all. The 2 ball end trick goes right out the window.

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No no and no

When I've gotten a new Strat I've often heard open strings buzzing in the nut. I had my technician install a second string tree and it eliminates the problem. Perhaps because I play brass nuts it's more of an issue.

Could be that they are ringing more. Metal on metal and all that.
As an aside. Do you find brass nuts wearing faster or slower than regular? Every brass saddle i've ever played needed smoothing and raising every 6 months odd. I'm wondering if thats because of material or a combination of material and being near the strike point.

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No no and no

I have never heard a string buzz in a nut hang up yes. The only way I could see that happening is if the string is vibrating past the nut up to the tuner in which case the guitar could have never been in tune from day one. Can you imagine how loose that string would have to be

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The strongest reason for the people to retain the right to keep and bear arms is, as a last resort, to protect themselves against tyranny in government.

Thomas Jefferson