Practice, Practice, Practice

They look great! Great detail.

Don't forget as you adjust your lighting you need to adjust your white balance. This might help with showing luster better.

Well this is silly, I hadn't been saving my pictures in raw format so there was a lot of stuff the editing program wouldn't even let me adjust. Even more to learn than I thought.

Okay, it goes something like this.

Choose lens and extensions based on whole coin vs detail shot. Make sure these are securely attached to camera.

Turn on camera.

Go into live shooting on the computer.

Adjust height of platform to get coin or detail roughly in frame and within range of fine tune focusing. Attempt to get coin mostly aligned properly.

Adjust lighting to get preliminary lighting.

Check to see that camera is level (mine likes to swivel).

Readjust lighting to minimize glare and shadows while preserving detail. This generally involves much experimenting with angles, bouncing light for indirect illumination, and diffusion. So far, diffusion hasn't been very successful for me, but I'll continue to experiment.

Play with exposure level and f-stop if it seems like the light placement is adequate but light level is off.

Recheck levelness of lens to subject.

Go into zoom mode on the computer if taking a picture of a whole coin face.

As carefully as possible, very slowly and very slightly adjust focus until I feel good about the sharpness. This usually takes a while because I need to rest my elbow on the table to stabilize my arm enough not to disturb the camera position too much while I turn the helicoid. However, because I'm resting my elbow on the table, it changes the distance from coin to lens a slight but noticeable bit (actually, it's extreme when using the Plan4), so I have to gingerly retreat from the camera and then go back to readjust again and again until it's where it really needs to be. This includes a final check to see that I haven't upset the lens level while adjusting focus.

Exit zoom mode, and as long as everything looks good in the live mode preview, finally snap a picture.

If it's a whole coin, I flip the coin, recenter and align it, check focus via zoom mode, make adjustments as necessary (generally not necessary), and snap the other pic.

If I was doing a detail shot, I may make extremely fine focus adjustments and take a bunch of photos to capture various depths of field so I can run them through the focus stacking program.

Adjust photos to the best of my ability in post.

I am with you, there's a lot of rechecking that I feel like should be more like "set it and forget it". I'm not sure what a good microscope stage would be as far as the upgrade, though.

My camera is a Canon Rebel EOS. The picture run through GIMP looks a little brighter or whiter or maybe yellower? than when I look at it in hand.

You have some "crushed blacks" that you can probably fix with camera settings. Since you're outputting jpeg, you need to ensure the full dynamic range of the RAW files is properly mapped to the jpeg image. You're probably already doing the EV compensation to eliminate over-exposure, which will cause you problems on the dark end of things. To fix that, reduce your contrast setting. This will brighten the darker areas, and allow you to recover those crushed blacks.

I still see some focusing softness on the left around LIBERTY. If you are taking care to level the camera, but still having these issues, you are probably at too large an aperture. For publishing at 1000x1000 as you are doing, you can tolerate a smaller aperture.

I am also not fond of the highlighting I'm seeing on the forehead. It appears you mitigated the forehead shadows by adding more lighting, or a reflector, but it is probably too much IMO. At least reduce this lighting so that the formerly shadowed areas are not highlighted.

Finally, the image looks a bit dark. You are not over-exposing on R, but you may need to compensate the overall image a bit more. Only you can tell if this matches the coin brightness well, but working in isolation I'd suggest adjusting the image to look like this: