expectations when applying for an adjunct position I applied for an adjunct lecturer position through a university's web site at the beginning of August.

I assumed I would be put into an 'adjunct pool' of some kind, or simply be declined . . . or something.

However, no one has reached out to me. I've logged into their application site several times since and my application is still active, supposedly. I have never taught before so I guess I'm not sure what my expectations should be. Is this normal? What should I be expecting?
In industry, hiring decisions are typically made very quickly.  When I was at Microsoft, if everyone on the interview loop liked a candidate, we tried to let them know before they even left the building that they should expect an offer.  We had to, because in industry, good candidates only stay on the market a few days before they get and accept good offers elsewhere.

But it's a completely different story in academia where hiring decisions often take months.  It's likely your application will have to be put to a faculty vote before they can make an appointment, even for an adjunct position, and often the votes are batched.  You say you applied in August, but Fall quarter probably didn't start until September.  And even after everyone's back for the start of the new academic year, it could be several weeks before their first department-wide faculty meetings.

Also, realistically, adjuncts are bottom of the heap at a university.  You're part-time hired help brought in as cheap labor to cover classes when they need you and ignored when they don't.  So deciding what to do with you is unlikely to be a high priority.

Bottom line is that what you're experiencing is pretty much what you should expect.  Sorry to deliver the bad news.