1) Would someone please confirm or amend this list of 3 types of math required to get a better understanding? The word "probably" in the quote above just made me wonder if the list could use some refinement (no offense to the original poster).

Yes, you'll definitely need all of those three things. That's the bare minimum you need to learn QM in depth. There is more math that you'll run into, but you can generally learn it along the way. The three listed there are things that you want to have a decent understand of before you even begin.

2) I would like to attempt to do this on my own without enrolling in classes. Would someone please recommend the order in which one might approach learning these new types of math, assuming that one is unfortunately starting only with a grounding in "Calculus I" (US term for entry-level class including things like basic limits and derivatives). Specifically, I'm interested in if there's a good, logical order in which to move from simpler math into these higher level types, while - in parallel - studying the practical applications of each in order to make progress on the physics side of things.

The standard calculus sequence is differential calculus, integral calculus, and multivariable calculus, in that order. Then linear algebra and differential equations. The order there is not so important, as long as they both come after calculus.

Also you should post this question in /r/AskPhysics, and/or the weekly career and education thread on /r/Physics.