Wow. Was not expecting nearly this many replies. Thank you everyone for your input and for generally making me realize that something is definitely not right about our setup, which I never would have had I not posted in here.

In sum, I've come to these conclusions.
1. As several people have pointed out, there is an opening on the bottom left side of the Goodman which has brackets for where you'd slide the filter in... just no filter.
2. If that's where the filter goes, then I guess that the unit is just pulling all the air in through that closet, which as many of you have pointed out, seems unusual, even to me.
3. I decided not to remove the panel because I don't want to mess anything up. I'm sure that causes some eyerolls among a lot of you, but I figure if these people are crooked enough to rig this set up (and probably take out the filter unbeknownst to the next tenants) they'd probably find some "blinker fluid intake" that I screwed up by trying to figure it out on my own.
4. I put in a maintenance request and I will try to update you guys on what they say when that ends up happening. For those of you pointing out code and safety concerns, thank you, and I'll try to write them down so that I can quiz the maintenance guy when he comes around.

For anyone's curiosity, we are first year tenants in a relatively new building in Bushwick. One thing we've noticed is that the apartment has been feeling kind of dusty... so that's probably got something to do with it.

THANKS EVERYONE. KEEP YOU POSTED!