How are skill check modifiers prioritized and ordered in Arkham Horror?
With all due respect to the amount of time you put in making that list, I think you may be asking the wrong question.

First of all, there's generally no fixed order or prioritization which requires a flowchart. You add up all the things that affect the modifier, and addition is commutative. If you have things that let you reroll, they can be used in any order you want. That is, the "flowchart" would be:


Add up all modifiers
Roll that many dice and see if you passed
If you failed, consider your options for re-rolling, re-roll if desired/possible, see if you passed, and repeat.


The bulk of your question, then, is just how to make sure you got the modifier right, and perhaps how to make sure you didn't forget a way you can re-roll. And I would argue that your list is not going to help you there, because your list is longer than the number of cards typically in play. It will be faster (and just as reliable) to simply look through everything in play every time a skill check is made than to consult a list like the one you're proposing.

Remember, there are relatively few global cards in play - perhaps an environment, perhaps the ancient one affects skill checks, and other things are very uncommon. Everything else is either on your investigator sheet or the little cards (items + skills + allies) right in front of you, and you can very quickly scan through those, until you start remembering that your shotgun always doubles sixes, and that your dark cloak doesn't affect luck checks.

If you want, you could try to use counters/dice/sliders to remember all the global modifiers (e.g. when the Heat Wave environment appears, set global indicators for fight to -1 and lore to +1), but that gives you yet another thing to remember to do (adjust those indicators when environments change) so honestly, you might as well just look at the environment card every time if you're having trouble remembering. Same goes for everything else - you could perhaps keep track of your current net stats, but it's easier said than done.

So, in general, just try to get all the players to keep an eye on everything, and probably someone will notice if you slip up. Keep your cards in consistent places: environments always in the same place, organize your items into combat and non-combat items. And don't worry so much - if you realize after the fact, you can probably either fix it or just let it slide.