Should I list this firm if it no longer exists and my old boss isn't practicing law anymore?
You are required to list it. If your state bar uses the NCBE form, you'll be asked whether each employer is now defunct; I would expect all non-NCBE states to solicit this info in some similar fashion.
Even if your answer is yes, you must still provide the name and contact info of your supervisor. If you cannot provide it (e.g., you can't remember the name), you must provide the name and contact info of some other reference who can verify that part of your work history.
Something similar to your situation happened to me. Many years ago, I had worked part-time for a tiny now-defunct firm for only a few months; by the time I was filling out my NCBE work history, I could barely remember the name of one of the firm owners. So I spent hours and hours trying to track down valid contact info. After submitting all the research I had found along with W-2 forms, which NCBE then had to try to verify (and couldn't, because the old addresses were no longer valid), they still asked me to submit a verifying reference. Lucky for me, by a stroke of luck I ended up finding a firm owner's valid phone number in an obscure online people directory. But the ordeal still held up my application for at least a few weeks.
My question is: when the bar asks me to list prior employment for the purposes of admission after passing the bar exam, will it weigh against me that my first job was for this firm?
No.
When I was in the middle of bar application & exam prep, I had scoured the web looking for all official or credible intel on what NCBE and state bars look for when they vet Character and Fitness applications (e.g., on
https://thebarexaminer.org/). I never came across anything to suggest that they would hold the fact of this first job of yours against you.
My conclusion at the end of all that research -- which I admit may very well be deeply flawed and shouldn't be taken as gospel -- is that bar examiners care about stuff like dishonesty (e.g., like fudging a problematic work history), crimes that reflect badly on your honesty and integrity, substance abuse and other personal circumstances that strongly suggest an inability for high professionalism and/or disregard for the law (e.g., 50 parking tickets in the past year), and serious financial problems. And by the applicant, not people associated with the applicant due to past jobs.
One thing to flag though is that state bars will contact your former supervisor and ask them to verify the info you submitted and comment on your character. One of my former supervisors (who loves me) showed me the 1-page letter he received from NCBE. It lists your name and any aliases, your position/job title, supervisor or associate name (whoever received the letter), and dates of employment; then it asks whether all that info is correct.
The verification letter also asked why/how the position ended, whether the position was full time or part time, whether the reference would rehire you (and if the answer is no, explain), and whether the reference believes you possess the character and fitness necessary for the practice of law.
That said, even if your former employer badmouths you, I think you're fine. Just give a brief, matter-of-fact explanation of how your employer became defunct. That those partners were disbarred (long after you left) would undermine their own credibility if they were to talk shit about you.
If I were in your shoes, TBH my biggest concern would be that they don't respond at all to the bar's request for info, thereby delaying your application.