Not an admin, but I can answer some of these questions:
1. Is there a problem with the forums? It seems to be having a lot of issues lately, taking forever to load, triggering virus alerts etc. Can we expect to see a permanent shutdown?
I think this is due to neglect not planned obsolescence.
2. Somewhat relating to the former, I have been running a fairly long term PBP here, is there a way to download those posts for my own personal review, preferably as a complete thread?? It is a lot to try and dig through to even find, let alone look through for some continuity question.
At the top of the thread there's a "Print" button. That will create a single page with the entire contents of the thread. You can print that, save it as a PDF (if you've the software for that), Copy and Paste it to a text document, or save the HTML locally. Plus a myriad of other obscure methods for cataloging the data
3. If I were to keep some kind of Notepad file with the web addresses for the specific forums I mentioned above will the addresses change on their own as the forum get older? (This might be a stupid tech question, but I honestly don't know, so please be gentle)
The URL (address) is static and won't change unless there is a significant system overhaul that changes the structure of the URL (ie, changing the forum software or something along those lines), and nothing is done to preserve the old content. So, are they perma forever? Well, kinda. At the very least you can find them in Google Cache, the WayBackMachine or the Internet Archives even if this server does disappear one day.
Hope that helps. FWIW, if you are looking to preserve your PbP in perpetuity, I'd recommend periodically saving the HTML from the "Print" button every week or so and that will allow you to just put that file up on another web server at some point to display.