So the filter is in that short brown box between the furnace and the duct that goes all the way to the floor?

It probably should be pointed left. It goes like this: Air Return/Intake -> filter -> furnace with blower -> ductwork to individual rooms. You typically do not have filters on the output, only the intake. The duct that goes all the way down to the ground is almost certainly the return air duct. The arrow on the filter is the direction of air flow.

Easy way to tell is when the furnace is running (or you manually set it to fan-only mode) you open up the tray, take out the filter, and stick your hand in there. You'll be able to feel the air flow. You need the filter both for air quality reasons and to protect the blower from grit in all the wrong places, but running it for a minute or two without the filter won't cause a problem.

It is entirely plausible, nay, likely you have a combo furnace + cooling that uses the same air handler. If that's the case then you'd only have the one filter since it would use the same intake either way. See that black-wrapped pipe about 4/5ths of the way up and the copper pipe about 1/2 way up? Those are probably refrigerant lines for cooling. Easy way to tell is to set your thermostat in such a way that the AC turns on. Now check that furnace to see if the blower is on. If it is then there ya go.

Your AC unit almost certainly does not pull in outside air. The outdoor unit condenses the refrigerant and cools it off with the radiator-like structure and giant fan. That liquid refrigerant gets pumped to the indoor unit where it boils through expansion (thus cooling down) inside another radiator-like structure and the air handler blows air through the very cold radiator, thus cooling your house. Household units, even window units, typically do not bring in outside air.