Digitizing slides with a camera: still too much noise

Question

Third episode of my odyssey to digitize our entire family slides collection.
If you want to know all the details of the project, I invite you to check my two other posts but here is a quick recap of the situation.
I modified a slide projector to be able to digitize fast a big number of slides by taking photos of them with a Nikon D5000 using a Macro reverse ring.
I'm shooting both in RAW (NEF) and JPEG and both of these formats have visible noise.
I'm now near perfect image quality (at least to my appreciation) but there's one detail I'd like to fix: there's still too much noise on the photos in my point of view especially visible with blue sky or skin for exemple.
My camera's settings are as follow:

Aperture: manually set a couple of steps down from wide open (as advised in an answer of another question of mine)
Shutter speed: 1/400
ISO: 200
Exposure compensation: 0
Picture style: Neutral
White balance: custom (preset with no slide in front of the light)

I chose an ISO of 200 because that's the base ISO of the Nikon D5000. However I also tried with a 100 ISO and 1/200 Shutter speed but that gave no change whatsoever (at least not that I can see with my own eyes).
I don't know with which camera the slides have originally been shot so I'm not sure if I'm just shooting the noise that is already in the original shots or if the appearing noise has been added when shooting the slides with my camera.
Also, as advised in this answer I added several layers of tracing paper to best diffuse the light rays in all directions. The paper I bought is the best quality I could find with the least grain. So I'm wondering if the noise might come from this paper.

So here are my questions:

Do you think the noise is already in the original shots?
If not, do you think that could be improved?

If that could be improved do you think that could be solved with any of these changes:

Changing the LED used for projecting the slides (I bought a cheap cool white 50W LED on Ebay)
Changing the shooting settings (Aperture, shutter speed, ISO, exposure compensation)
Changing the camera to use one with a bigger sensor size (full frame or medium format)
Removing some tracing paper layers or change it completely to another kind of diffuser
Another idea ?

Some changes are acceptable to me like changing the settings or the diffuser type and some aren't like buying a full frame camera for exemple. But even if some of these changes aren't doable I still would like to know if you think they might solve this issue.
I really would like to get the best photo quality before digitizing thousands of them.
Thanks for your answers.

Tetsujin · Answer

Perhaps not a total fix, but some ideas in hopefully a helpful direction.
I doubt the noise is camera noise at 100 or 200 ISO, if you are completely filling the frame of the D5000. My D5500 is newer, but I'm almost sure it couldn't be that much noisier than mine.
It is, btw, just about fixable in Photoshop, pic at the end of a quick 'fix' attempt.
To fix your focussing issue, shift the camera to Live View, then use the + [magnifier] button bottom right to zoom in as far as you can. Adjust manual focus whilst you can see one of the hardest-edged details in your slide.
[I've subsequently seen the new question & don't know why the reversing ring can't focus… unless it has no electrical connections]
You have some vignetting [a darker ring round the edges] - possibly because your diffuser isn't absolutely 'perfect' or your light source is too close behind it, or indeed that the lens itself [not designed to work that way round] is vignetting. You may not be able to fix this entirely, but you could perhaps swap out the tissue I previously suggested for some proper diffuser gel. You can get this for just a few $£€ on eBay. It is, comparatively, optically perfect. It may still not be quite enough to get rid of all the vignetting. This can anyway be fixed in Photoshop to an extent.
You also have a purple-ish colour shift. this might be the film or it might be your white balance, or even the LED. Probably film. It may well change between film batches too.
With some Photoshop 'magic' you could actually work out how to automate all of this - but that's definitely for another question [& someone better at Ps batch scripting than me].
Anyway - a quick attempt to smooth the noise, pull back some of the purple, punch up the remainder, knock back some of the over-bright centre & balance out the vignette. [It's not perfect, it's a 5 minute quick fix.]

eBay UK link to an example gel diffuser (not a recommendation, per se)

Digitizing slides with a camera: still too much noise

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