How do you keep parts this tiny from going airborne?

The trick that solves most of this at once: never try to hold the finished part. Leave everything attached to a chunk of stock big enough to grip properly, do all your turning, grooving, and undercutting while it's still one piece with the bar, and part it off dead last. Even then, part off with extra length on the stub so there's something to grab for cleanup, and finish that stub end by holding it in a pin vise or a tiny split collet instead of the lathe chuck. Chasing a 0.6mm bit around a three jaw is how parts end up under the bench for two days, ask me how I know. For grooves and undercuts, grinding HSS isn't the wrong move, it's basicly the only move at that scale. Take a thin blank and grind it nearly to the width you need, then stone it the rest of the way. A shard of hacksaw blade also works for shallow grooves if you grind the teeth off one side and put an edge on it. Keep the tip dead sharp and take spring passes instead of cutting deep in one go, tiny parts deflect before you even feel it. On measuring, first check whether you actually need the microscope. A standard 0-1 inch mic goes down to zero and will measure 0.6mm no problem, so if yours bottoms out you've probably got a bigger frame mic and a small one might be worth picking up. A cheap USB microscope is handy for checking profiles and groove shapes, but the magnification lies about size unless you put a ruler or a gauge pin in the same frame as a reference. Calipers at this size are eyeball candy only. One more dumb trick that isn't dumb: turn a sacrificial test neck in the same setup first, measure that, and dial in your cross slide before touching the real part. Re-cutting a part that took twenty fiddly minutes to get this far is worse than spending two minutes on scrap.

Before all the fancy camera gear we used to check small work against gauge pins or a drill blank. Roll the part and the pin togehter on a surface plate, you can feel a difference smaller than you'd think. Cheap and it still works. Also, for the really fiddly bits I keep a stick of shellac on the shelf. Warm the part, dab a bit on a brass arbor, hold it to the work and let it set. Old clockmaker trick, held plenty of tiny things for me over the years and it lets go with a little heat when you're done. Curious what material you're cutting. Brass and aluminum forgive a lot at that size, stainless is a whole other game.

One thing that helped me with stock that small was feeding the bar all the way through the spindle and letting it stick out the back, so only an inch or so is working in front of the chuck. A 0.6mm bit of steel whipping around behind a three jaw is its own kind of nightmare. Also I mostly use free-cutting brass for this stuff now, it cuts so much nicer at that scale that I gave up arguing with mild steel. Curious what material you're on, and does that leg hair comparison actually help with scale or is it just for bragging rights?

That leg hair photo is humbling, mine came out about the same when I tried stuff this small. One thing I haven't seen mentioned yet is air movement. At that size a door opening or the compressor kicking on will send a part flying. I close the shop door and shut off the dust collector before I even part one off, and I keep a magnetized parts dish right under the chuck so anything that drops lands somewhere findable. Also, a ground-down sewing needle works as a tiny boring bar if you ever need to hollow one out.