Is this what my air intake should look like?

Hi, I live in a third floor, 1 bedroom apartment. My AC struggles to cool the apartment adequately, and our electric bill is quite high. We've had the maintenance guys out here a couple times last summer and a couple times this summer and both times they say there's nothing wrong with the a/c. I start poking around on the internet and trying to educate myself as much as possible, and I'm wondering if the reason it is not working properly is because it is not taking enough air in. I've taken some pictures to show you what I'm talking about.

this is the vent on the ceiling, has a filter behind, so i'm pretty sure thats the intake
this is when i open it and move the filter
this is to the side of the filter. basically if you stick your head in the vent then look to the side this is what you see.

so i guess this boils down to a couple things:
1. This IS my Intake vent right?
2. Is that ducting blocking the intake of air?
3. If it is being blocked, could that be causing my AC not to work properly?

 

Yes on all three question's.

 

So I consulted my fiance's father (certified ac guy, has worked on tons of apt complexes, ac units, etc.), and he says that my problem is the fur down coil in the bathroom. He described the unit in the ceiling of my bathroom to a t, and said the coil needs to be cleaned or replaced. I suppose this development might make this the wrong sub forum for the thread to be in, but does that sound right to anyone? maybe a combination of factors then?

 

I have never heard of a fur down coil, is this a typo? Could he have said fan coil?