In kind of urgent need of some advice from someone who's knowledgeable about unemployment insurance and/or employment law. Attorneys/other legal professionals are preferable given the nature of my situation.

LONG story below. Thanks!

I left my paralegal job at a giant law firm in July to do marketing for a smaller firm that then unceremoniously terminated me in September (long story, but basically I still don't know why they did -- and this isn't about that, though it is troubling and problematic in other ways). 
In the termination meeting, they explained that I'd be getting severance (yay-ish) and could use COBRA after the end of the severance period, but nothing about unemployment insurance or receiving a representative from the state of NY labor department assigned to my claim case -- the official termination letter said nothing about that either, as I confirmed yesterday, but I've spoken to a representative at the department of labor on the phone who says I was supposed to basically receive one immediately.
Within minutes of my termination - and up until today - I have been applying to jobs online (job boards and directly on company sites), through recruiters and other contacts, etc....all the steps necessary to qualify for unemployment insurance, EXCEPT for actually submitting a claim. I didn't know it existed, though, before a friend's roommate commented while I was doing my job search at their apartment, otherwise I would have. And, of course, the very next day I was offered an ongoing temporary part-time job that I still have. But because I'm still underemployed, I've still been filing the claims with the department of labor online pretty much every week (even though I'm usually making more than the determined amount to qualify).
I requested back-pay or for the department of labor to change the date of the claim to include the dates I was unemployed but not yet in the system, since I was looking for work but didn't know about unemployment insurance -- it was denied in a "final determination". 
So I requested a hearing and was given a hearing date the other day of next Wednesday afternoon. 
I'm not quite sure exactly what would be helpful to prepare with or how - or if the case is even winnable given my circumstances.  

• Is this something, for example, that I should also be worried about/suing/writing to the firm that terminated me about, considering they were supposed to tell me about UI but didn't? 
• Should I bring a copy of the termination letter from them, a printed copy of my job hunt chart (a color-coded, multi-tabbed google sheets document), and/or emails between my recruiters and me on the day of termination as proof of how quickly I sprang into action?
• The department of labor is claiming that I can't use the excuse of being busy applying and interviewing for jobs as a reason for not having submitted a claim in time to receive the benefits needed during those weeks...but that's not what I'm saying. I'm saying I didn't know because no one told me it was something a fired/unemployed person should do until weeks after my applying/interviewing had already begun. I think they misinterpreted or misread my emailed request, but unsure if the real request would make a difference to them properly interpreted anyway.

[Side note: I'm currently qualified for UI, I've confirmed with the call center person (though not at the full amount), so I know this isn't about any weeks since claiming but more my desire to make up for the dates that I didn't know UI existed and should have known.]