I don't have experience with Nikon cameras and haven't used their lenses since ~2006...but wanted to add on some info about your research:
70-200mm is the most commonly recommended range for close-range action.
70-200 is definitely a very popular sports range and it should serve you well. However, 15ft isn't that far away and if you are using an APS-C sensor, you may struggle to get a full body or contextual shot: 70mm at 15ft is just too much focal length.
The supplemental lens to the above tele-zoom is a wide-normal zoom like a 24-70 f/2.8. These are, however, expensive. You can supplement your tele-zoom by using a 50mm f/1.8 or 35mm f/1.8 - both on the cheaper to acquire side of the fence.
f/2.8 will allow for low light situations, inherent in indoor events or cloudy days.
Kinda-sorta. f/2.8 is known because it's the fastest many pro zooms can go (16-35 f/2.8, 24-70 f/2.8, 70-200 f/2.8). But, indoors is incredibly dim. Really - your eyes say it's okay while your camera cries out in partial blindness.
With how good High ISO noise control is these days, you can get away with an f/2.8 lens and using a higher ISO to maintain fast shutter speeds. I would always, however, recommend having a cheap and fast prime in the bag, just in case (see above point on a 50 f/1.8 or f/1.4...or a 35mm or 85mm sub f/2 lens). Keep in mind that the difference between f/2.8 and f/1.4 is 4x as much light - so if you're ideal exposure says you need 1/125 at f/2.8...that is equivalent to 1/500 at f/1.4!
An auto-focus which works well with the base is necessary to maximise fast+clear shots.
Good autofocus and autofocus tracking helps, but this one is also very skill dependent. Learn about "Back Button Focus", Focus Lock, Exposure Lock, and read up on how to use your camera for sports shooting for tips and tricks to get the most out of it.
Remember, people took some awesome sports photos long before autofocus was a thing. Autofocus will definitely help you get more keepers, but it's no full replacement for skill.
At sports speeds stabilisation is less important, especially with a tripod. Vibrations from the floor could be minimised with a better tripod or strategic placement.
At sport speeds, you probably won't be using any sort of VR/IS. But, it comes in handy for not-quite sport speeds. Say you're shooting at 200mm and 1/125 second. That's a tad slow for handholding at 200mm (you'd normally want 1/200 or higher), and you'll probably get some action motion blur in your shot. But you want this for this shot. The VR/IS will help make sure you don't get camera shake in your shot.
So, it does come in handy. Yes, you can live without it. But, if it's an option on the table, then go for it!
Also, don't use a tripod for sports. You can't move quickly and your rig can be a tripping hazard for others. If you need the extra support, get a monopod.
An FX lens will have more longevity in a photographer’s toolkit because as skill and funds improve to full-frame the lens will still work well.
There is a lot of info on "upgrading" to full frame out there - much of it reading like a full frame is the holy grail. It's just another tool. It's got some pros over APS-C and cons against APS-C. Moving to full frame may not actually be in your future.
Think long and hard about this one.
Should I rent lenses?
I know you didn't ask this, but you should have. And the answer is yes. Yes you should. Find a local shop or website (lensrentals.com for example) and try out every lens you think you want before you buy it. Better to spend $50 taking a lens for a test run to a few matches and realizing that it doesn't work for you than buying at full retail only to have to try to sell it secondhand or stare at it taking up space in your bag.