I can understand anxiety coming into this, but take a few deep breaths and try to not let it get to you - it's all very clear, straightforward, and your careers councillor can help you through the whole thing.
I also have documentation set up from various doctors/specialists courtesy of my mother.
You may have to see them again - to memory, they might have needed to sign specific forms. There is definitely a coversheet you need to send it in with. Alternatively, you can get your doctors/specialists to sign something online about it all, which means you won't have to send anything in.
Is is merely a means of improving your chances for a certain course (even though it does explicitly reference that "there are no guarantees with SEAS due to the competitive environment")? Or, can it actually enhance your ATAR slightly, as indicated by the Monash SEAS calculator?
From what I'm aware of the system, your ATAR won't actually increase, just that you'll be eligible for courses with a higher ATAR depending on the "severity" of your issues. (severity used extremely loosely - issues are issues, not really fair to give them a numerical value of how "bad" they are)
How long does this take? Do I have to apply for different universities? Do I have to write a huge thesis on my hardship which I don't necessarily want to revisit in the slightest?
This is the beauty of VTAC - it is a one size fits all (except I think for scholarships, you have to apply to some unis directly, but VTAC warns you about this a bajillion times, so you won't miss it) system. You won't get told of if you were successful or not, or how much you'll be getting out of it (unless I just missed that message last year...).
You will have to write about your hardship, but given the nature of it, you might be alright if you have your specialist detail the whole thing for you - they can then include in this write-up that you did not give them information because of emotional trauma. However, if you can, it would be best if you could write something light about it in your own statement (as everything you apply for, you'll need to give your own statement as well as a specialists).