That is a task I would not take on. A modestly damaged image could take several hours to restore and correct, and even longer if you need to scan it.

Photoshop is an outstanding platform to use for photo restoration.

This website - https://www.retouchpro.com/ - is a trove of information.

The image below shows what I was given - a scan of a water damaged image - and the result after about hour of work.

Retouching requires OCD-level attention to detail, and a comprehensive familiarity with the tools and techniques available in Photoshop. The good thing is that once you get good at it, you can make decent money doing this. Good retouchers are few and far between. This was a relatively easy retouch.

D850 and D750 are both full frame cameras. To shoot sports with full frame generally requires longer focal length lenses, which are necessarily bigger, heavier and typically a lot more expensive.

If you want to avoid that, stick with APS-C cameras like your D5100 (16MP, 11-point AF, 4 fps).

The very best upgrade that keeps you shooting Nikon DSLRs would be the D500 (21MP, 153-point AF, 10 fps).

A close second place would be the D7200, which has been discontinued and is now only avail. used. (24MP, 51-point AF, 6 fps).

I'd rank the current D7500 in third place, because in some ways it's an upgrade from D7200, but in other ways it's a downgrade (21MP, 51-point AF, 8 fps).

The Z50 probably has better battery life than you've heard (standardized CIPA tests require cameras with built-in flashes use them for half the test shots, which heavily drains the battery). Still, as mirrorless cameras go the Z50 is just an early attempt from Nikon and is actually getting a little dated. If you want to stick with Nikon, wait to make the switch to mirrorless. I suggest you "make do" with one of the three Nikon DSLRs for the time being. All three of them would be very solid upgrades from your current camera, can use any and all lenses you already have (some older 3rd party lenses may not work). But eventually you will probably want to move to a mirrorless system, and no doubt Nikon will be introducing new and better ones in the future. The upcoming Z9 sounds like one heck of a camera, but isn't what you're looking for by any means.

All these cameras can handle everything else you say you shoot... Sports is among the most challenging. It demands highly capable AF systems (and photographers who know how to use them). Faster frame rates are "nice"... but not as important as AF speed of acquisition and ability to track movement. Often with sports it's also necessary to use a faster shutter speed, which means increasing ISO and these newer cameras are no doubt better at that, too, than your current camera.

Here's a comparison of D5100 vs D7200:

https://cameradecision.com/compare/Nikon-D7200-vs-Nikon-D5100

And here's a comparison of D500 vs D7500:

https://cameradecision.com/compare/Nikon-D7500-vs-Nikon-D500

Those comparisons can be changed if you wish. And they just summarize key differences. You should search for more detailed reviews of any model you want to consider.

Just for example, I prefer to use a battery grip on my cameras, both to double the battery capacity and for the vertical grip and controls it provides. The D500 and D7200 both can be fitted with a grip. The D7500 cannot. For some reason Nikon chose to design it without that as an option.

Another example: "Flicker Reduction" is a feature in the D7500 and D500, but not fully implemented in the older D7200. If you anticipate shooting sports under the lights, this may be important. This is a feature that greatly reduces exposure problems that occur when shooting "under the lights". Some types of lighting (e.g., fluorescent) rapidly cycle on and off. It's so fast our eyes don't see it. But the short duration of our camera's shutters can occur at the wrong time during the light cycle and result in images with serious under-exposure and color shift. Before "flicker reduction", we had to use a longer exposure to avoid this. But that's not a option when shooting sports, so I'd see a lot of badly exposed images when I was shooting under certain types of lights. As many as half my images could be bad enough that they were worthless, so I shot a lot of extras. "Flicker reduction" for still images solves the problem by detecting the cycle of the lighting and timing the shutter release to the peak output of the lights. Usually this is so fast you don't notice, but every once in a while you might have a slight shutter delay. This is not to be confused with flicker reduction in video, which many cameras had long before a similar feature was available for still photos. The D7200 has flicker reduction in Live View and video mode.... neither of which help with most sports photography.

D7200... $434 to $549 used (depending upon condition, shutter count, MPB.com)
D7500... $997 new, used from $749 (MPB.com)
D500... $1597 new, used from $789 (MPB.com)