TBonz

I would not consider the 55-300 good for sports due to the maximum aperture. If you have decent light, it might do OK for what you want, however inside you will likely have to increase the ISO to an unacceptable level to be able to get a reasonable exposure.  

Either a zoom or a prime lens with a fixed aperture (as opposed to the variable aperture on the liens you asked about) would be a better choice. 

This isn't to suggest you can't take some sports shots (indoor or outdoor) with a D3200 and a 55-300.  However, that may not be true of every venue you want to shoot in.  Even if you don't get the reach you would like, something with a better aperture might allow you to get images in places the lens you asked about would not allow.  I would want to be sure I was able to capture images first.  

I think I would choose another option, but hard to say what I would choose.  To make it easier, I would choose a full frame body like the D750 and a nice portrait lens,  Either the body, the lens, or both can be found much less expensive if you purchase a lightly used model.  You also might want to consider a zoom lens that includes the "portrait lens" length so that you will have greater flexibility for shooting other subjects.  

Using what you just said, I will provide a thought and an example.  First, there is nothing wrong with sticking to letting the camera do the work, but there will be situations where that isn't an option if you want to get a good photo.

Thought - With any type of action photography, you will have failures.  Don't let those be a problem, but always work to reduce them and try to learn from them.  The most important step in that process is practicing with the camera and seeing what happens when the same shot is taken with different settings - BEFORE shooting a once a year event. Even just taking photos of cars driving by will give you some practice and examples to learn from.  There are 3 settings ISO, Aperture and Shutter Speed and they are entwined together.  Do you want to stop action or show movement?  The shutter speed controls that but the faster you shoot, the wider your aperture will need to be for allowing light in and that impacts your depth of field.  Raising the ISO can make the sensor more sensitive to light but that can also introduce grain in the images.  

Example, shooting exclusively in shutter priority basically allows you to choose a shutter speed, but lets the camera decide the other settings.  Technically you can do the same thing by using the camera's meter and manually make those settings.  I am not suggesting you do that - just that you CAN do that.  For your old camera, it was choosing settings just as if you were set in Program or Auto but with some added information - that you wanted it to make the Sports setting adjustment.  I expect that is documented somewhere in the manual or can be determined by looking at the EXIF information on photos taken using that setting.  You can use that information when shooting Shutter Priority or Aperture Priority or Manual to give you a starting place in similar situations.  From there you can use the camera's meter and information from test photos to get you set without the sports setting.

Use both...Both Peter and I have DX and FX bodies.  The most important thing you can do is to get comfortable with your equipment.  While each of the bodies has features that make them more desirable for a particular style of shooting, the other body is still capable of shooting the same thing.

It sounds like you are more worried about you having to put some learning in and not be able to rely on the camera to do the work for you.  We have both gone through that process as well - and we still do at times if we are shooting anything unfamiliar.  I enjoyed that process of learning and it is why I am still shooting this many years down the road.  I expect Peter did as well.  You have the benefit of learning with digital media so you can immediately see the results and adjust your settings with test images before you "start".  You can even use the EXIF information (ISO, aperture, shutter speed) from your current images that you like as a starting place.

You can get excellent results with both of those cameras if you follow some of the basic "rules" of photography and trust yourself.  If you just want program modes to pick what the camera thinks are the best exposure settings, then you won't need to put in the learning and you likely will not get the best possible results.  Some folks want the image without the effort and that is completely OK.  That is why Nikon and other manufacturers make a variety of cameras. 

Do not think of it as the smaller the diameter, the larger the #.  Instead, think of it the opposite way...the larger the number, the larger the depth of field...

Other than that, see the answer above

MSPaint is not what most would consider a photo editor even though technically you can use it to modify a photo.   Sounds like gimp has the same issue if it is giving you similar results.  Shooting at a lower resolution should not provide a better image.  I think you need to try Nikon's free tools or some other photo editor...

If she is photographing sports, the 24-200 or any Superzoom would not be the best choice.  Certainly the Nikon Z 70-200 would be the best choice in that range.  I expect that the OP would have better luck tracking down a used F model 70-200 f2.8 VRII which should work well with the adapter.  Perhaps one of our folks has some experience with that combination.  I don't have mirrorless, so I am speaking of DSLR bodies right now. I don't recall anyone I've worked with / around shooting the Sigma 70-200 however I know several who shoot the Tamron.  I thought the Tamron was slower focusing than the Nikon VRII - not by much but it doesn't take much to miss "the" shot with sports.

“It has been a source of great pain to me to have met with so many among [my] opponents who had not the liberality to distinguish between political and social opposition; who transferred at once to the person, the hatred they bore to his political opinions.” —Thomas Jefferson (1808)

I don't have a Z6, but...

You mention it hunting for focus.  I prefer to shoot with a singe or small focus area.  You may also want to search for threads on Back Button Focus.  I am not sure how large the buffer is on the Z6, but sounds more like a slow or damaged card.  Did you format the card in camera?  If not, you should. 

Hopefully that will help...

Storage is cheap these days.  Why limit your abilities worrying about it?  As someone who is more of a photojournalist than an art photographer, I don't normally choose to remove items from an image, but even then, some post-processing, especially from a RAW file, can restore an image to more effectively convey what you saw without the image looking over processed.  Being able to modify images can take a poorly exposed image and turn it into a usable image although you should strive to capture the best image possible when shooting.  

I don't know the answer to that but it isn't like it s bank account information.  I have never worried about it and it could come in handy if you needed to show the image was one you shot vs. removing the info.  

The materials were not sub-standard but standard for the day...I have seen the same problem on cameras from other manufacturers.  I remember a medium format a friend had that had a similar issue.  I inherited a 35mm that also had the issue.  I believe there is a thread here on the site where one of the members removed the failing cover and replaced it with a leather cover.  Bottom line is that if you don't like it, you can fix it or sell it or donate it.  While you can also throw it away, I would urge you not to as I am sure someone will take it...

Several things can cause an inability to focus, most likely a lack of contrast in the focus location.  It could also be that the camera could not adjust the other exposure settings based on the aperture you chose.  Lastly, if you were using the internal flash or had an external flash attached, the camera won't fire unless the flash is charged and ready to go.

Personally I would choose the D500 over the D750.  If you go to an FX body, you will need to replace your current lenses.  Yes, you can use them, but I don't expect you will like the results.

The XQD card is the faster card, so most people would prefer to send their larger RAW files to the XQD and their smaller JPEGs to the slower SD card which is one reason Nikon Shooter said what he did.  I don't see any benefit either.