Let's simplify the fuel supply for a moment and assume no check valves. Fuel line from the tank, into a pump, out of pump and into the carb inlet, through the seat - flow being metered by the needle and float, then into the carbs' fuel bowl. The seat/needle/float controls how much fuel gets into the carb. The fuel pump is designed to output a certain volumn of gas at a certain max pressure, and the carbs' seat/needle/float set up is also designed to operate at a specific pressure. The S/N/F assembly works just like a toilet; as the water in the tank rises so too does the float. Once the float gets high enough it closes the inlet valve and the tank stops filling.
Lets say the carbs S/N/F assembly can only control a max of 7 PSI (which probably is pretty close, it varies between different brands/models of carbs), but the pump is putting out 10 PSI. The additional PSI is going to over come the S/N/F and push fuel into the carb bowl right past the needle. The float is trying to keep the needle on the seat - stopping the fuel, but the excess FP is pushing the needle off it's seat - flooding the carb.
Now toss in some check valves. Conceptually, they could be just about anywhere in the system but their purpose is the same; prevent fuel from flowing backwards, thusly keeping the lines and carb full of gas, allowing for quick/easy re-starts. I don't think you have a check valve problem.
Symptom 1: Gas flows when you replace an inline filter. Are you certain it is under "pressure" or simply following the laws of physics? If you can easily stop the flow with your finger it probably isn't under pressure. Is this engines' fuel line physically lower than the other, or lower than the fuel tank? Is your fuel system designed with automatic cut off valves?
Symptom 2: Engine starts dirty. Could be a few different things really; ignition system is weak casuing excessive cranking - which causes too much fuel into the engine before it finally fires, the carbs S/N/F assembly is weak allowing too much gas into the carb flooding it.
So, a couple of checks. If the fuel delivery system has auto cut off valves, and if they are designed to close when the ignition is turned off, ensure both are in fact closing. Perform a fuel pump pressure test. Ensure the ignition system is operating properly. Finally, if the carb is old and has never been apart, might have to disassemble it and replace the S/N/F assembly.
Fun stuff, no?