Auto white balance is a bit risky with focus stacking. Your camera may decide between shots a different color temperature, ruining consistency. It usually does work reasonably well, just saying it's better to take control.

My approach is to set the LED lights at 5600K, yet 6000K should be close enough. Note that I shoot in RAW, allowing me to finetune WB afterwards.

I'm afraid I can't help with Zerene Stacker, I use Helicon Focus. Mostly because it supports RAW and is also very fast.

Hmmm, hard to say why you're not getting sharp images. 1/125 should usually be fast enough. Are you using silent photography? A mechanical shutter is a big source of vibration. Check in live view if your subject is vibrating. If it is not vibrating there, likely the shutter causes it.

Yes, all compound eyes dry out, best thing is to process as fresh specimen as soon as you can. If not possible, keep it in the freezer. Before using it, consider contact lens cleaner to soften/clean it again.

Dirt is complicated. If it's loose dirt, just use warmish water with dish soap and gently rock the specimen, the dirt will fall off. If the dirt is sticky, you need additional chemicals.

Hot spots are indeed best minimized using diffusion. For very shiny surfaces, this still may not be enough. Don't really have a solution for it. Allan has a video on how he uses polarized light to block reflections.

For step size, I use this:

https://www.wemacro.com/?p=529

Next, in Helicon Remote I set the start and end point, the program calculates the amount of steps.

Hope this helps!