If I calculate the VO2max on each split of a run using the formula from here. Would the VO2max estimate for the entire run be the mean of all the splits?
V02max is more of an absolute number. Think of it like testing your max bench, or max vertical jump. It's the maximum amount of oxygen you can uptake at the time of testing. Something you can track over the course of a phase of programming, but you won't see much change in it, even from run to run. It'd be negligible from split to split. Not really worth it here.
It can be increased (just like jump height or strength) but you're more likely to see it change based on a test say one to three months apart. And even more likely to see it change if you're using training techniques designed to increase it.
It's more of a performance marker or an indicator of potential. It's best tested with a metabolic cart and most estimates/formulas aren't that great generally speaking. That doesn't mean you can't track it with a formula from phase to phase of your training program, but tracking it from split to split or run to run is probably too frequently.
It's only one of many performance markers you may want to track.
No, those formulas are for the VO2 amount you used/expended and is not your max. The largest one of the set should be a good minimum number for what your max might be. The max VO2 can be considered a physiological constant of your body over an entire run, and it only changes over a matter of months of running or at 2 or more weeks at best. I dont know if you can achieve your max VO2 during exercise. I believe that you cant achieve your max heart rate without fainting, so I bet this is similar.
There are alot of calculators and your link has a link to one, but check out some places locally that can do it. On their website usually your find the method or procedure they use summarized. That can help you understand what exactly the VO2 max is.
Also note that if you want to predict race times you would need to know your lactate threshold instead any races or efforts that last longer than 5-8 minutes.