Mike,
As always, Charles' analyses and recommendations are spot on.
For the compound microscope, another option I do not see mentioned is to just remove the eyepiece from the trinoc tube and let the objective form its image directly on the camera's sensor. Optically, this would be like having objective on bellows. It would be even better in this case if you could get the camera mounted closer to the end of the trinoc tube. With some awkwardness, it is probably possible to mount the camera on tripod slightly above the end of the tube, then light seal around the gap with some cardboard or dark fabric.
At present, your adapter with eyepiece is giving about 2.6X magnification on top of the objective's rated magnification. For example, using your 4X objective, 1 mm on the ruler spans about 10.3 mm on your sensor. 10.3mm/1mm = 10.3X, 10.3/4 is about 2.6 .
If you could mount the camera within an inch or two of the top of the trinoc tube, with no eyepiece, then you could cut the additional magnification to perhaps 1.3X. This would give you a brighter image as well as a larger field. With the 4X and 10X objectives, it may also improve your image quality (depending on what the eyepiece currently is doing). With the 40X, image quality will probably degrade because the added "tube length" from objective to image will introduce aberrations.
So, the idea behind using flash to reduce vibration is that the flash doesn't go off immediately upon opening the shutter. Does the camera do that automatically or does it have to be set that way?
The main advantage of using flash is that flash exposures are very short, typically 1/1000 second or less. This has the usual effect of "freezing" motion because the image cannot move very much during the brief time that the light is on. So flash does not really reduce vibration, it just reduces the bad effects of vibration. With most cameras, the flash
does trigger immediately after the shutter opens. Some cameras can be set to trigger the flash just before the shutter closes. This is called "second-curtain sync". I don't know whether your camera provides that.
So, low ISO and long exposure yields better quality than high ISO and shorter exposure?
Yes. Each pixel is essentially a photon counter, with the maximum count determined by the ISO setting. Low ISO sets a large maximum count; high ISO sets a smaller maximum count and compensates by scaling. Noise in the image is mostly due to sampling uncertainty in the counts. Larger counts have relatively smaller uncertainty, so lower ISO gives less noise.
There is also some light-mimicking thermal noise that accumulates even in the absence of real light. To minimize this noise, you want bright light so that maximum counts can be accumulated in the shortest possible exposure time. This is another advantage of using flash.
--Rik