I know the big grey one is an inline fuel filter. But is the other one, connected to the carb, a fuel filter as well? If so, can I remove the big grey one?

 

Yes, the item circled is the factory fuel filter. They don't need much crud in them before they get clogged up. This is why many people will add a bigger one, like your pic shows.
I will say this though, that is a bad location for a filter on a 300, since the exhaust manifold is directly below it. My truck had one of those clear type in the same location, and after driving it, I could shut it off and pop the hood to watch the fuel inside it "boiling" from the heat.
I relocated the big filter to the line between the frame and fuel pump, but left the filter in the carb. Haven't had any more issues since relocating the bigger filter.

 

Nice. Thanks Rogue.

 

my truck has two fuel filters as well idk why some truck i see run just one but hey the cleaner the fuel! lol speaking of fuel filter i needa change mine

 

As a kid I was told to always install one filter b/f the pump, and one after. I have always done that.

 

Here's what I've done and the why:

I installed a new filter at the carb. I also installed a larger plastic WIX brand filter in the line just before the fuel pump (just as F250 advises).
The filter in the carb wears out the threaded hole a little every time you change it, so I rarely do; instead, I change the big plastic filter by the fuel pump regularly. This filter also keeps crud out of the pump.

 

I agree with the pre/post pump filter suggestion. A filter before the pump keeps crud out of the pump, and it's small but very important check valve. Another filter as close to the carb as possible will keep crud from getting into the carb, and it's small but very important check valve.
The crud could be dirt, or just little bits of rust or rubber from the fuel lines.
Best part of having 2 filters in line, if one plugs up on you in the middle of nowhere, just delete it and you still have the one at the carb as a last ditch safety net.