Image quality is not in the same league as some soldering videos I see on Youtube, that I think were shot with normal DSLR cameras or camcorders.
Even with a really good stereo microscope you will be hard pressed to match a DSLR with a good macro lens at low magnifications. Your Greenough microscope has to fit two objectives in the same space and, compared to the camera's, those objectives are just ridiculously simple, two-element achromats. The Olympus
XA pocket camera, from roughly when the archetype of your microscope was designed, had a fixed, 35mm lens with three times as many elements!
That said the video you should get, with good quality optics from the microscope photo port to the camera, should be much closer to what you see through the binocular eyepieces and suffer from nothing like the chromatic aberrations and barrel distortion you see in your video.
Image quality degrades towards the edges, especially noticable on the camera port.
Even a high quality two-element achromat objective can only be expected to be flat and sharp, at the eyepieces, over the inner 60% or so of the field of view, that is ca. 0.8 x R, if R is the radius of the field of view. The microscope would need semi-plan or plan objectives, each with three to five elements, to be flat and sharp, respectively, over roughly 80% or 95%+ of the field of view.
All that is through the eyepieces. The camera has the added challenge of having a flat sensor that is stuck at the same focus from the center out to the edges. The objective's curved optics project a curved image on to the camera sensor. If the center is in focus, the edges will be at the wrong distance, or vice-versa. Semi-plan and plan objectives, reduce the problem, in that they project an image that is flatter out to a greater radius.
Shallow depth of field on the camera port.
Unlike a pair of human eyeballs, the camera can not change its focus during an exposure and meld what it sees with the mind's eye.
Focus does not seem very even across the board at least when using 0.7x barlow lens.
Getting good images with a barlow lens can be a challenge. The lens may not exactly match your microscope's objectives and it may be slightly out of position if it screws in or clamps on. The objectives also look through the sides of the lens and not through its well corrected center.
Have to manually adjust focus when changing zoom level which is awkward esp. because camera has shallower depth of field and hard to predict whether "good enough" focus on the eyepieces is also good enough on the camera port (and because I don't have a boom stand and all adjustments are wobbly.)
Now that is something I think you should try to fix: it may be causing the issues you see in the video and, personally, it would drive me to distraction. First, do the song and dance to get the eyepieces parfocal. Then immediately after the last step, don't touch the focus or zoom, and try to adjust the camera port and relay lens to get the camera in sharp focus so that it tracks the eyepieces'.
Can't easily switch objective lens e.g. casually add/remove barlow lens while working (Mantis seems to have a fast switcher.)
It is a bit of a hack and may not work that well near maximum zoom, but some folks swap in a pair of 20X eyepieces when they need a quick, closer look.
The 3D depth does not seem especially vivid to me.
At high zoom levels or at all of them?