OK this is the order...gasket metering diaphragm (on the carb) , diaphragm assem'y metering.
Soft diaphragm fuel pump, gasket circuit plate, harder gasket.
The boot is fine.
I just stumbled into this discussion, so let me complicate it some more.
First lets discuss cleaning for current production carbs. In the US it's all about EPA, and even though you are in UK you get the same carb materials on a MS 361 as I do. All these new carbs are way different than they were even 10 years ago. The materials are more delicate to harsh cleaners, and the pump diaphragm materials are made out of MYLAR and other alcohol resistant materials and are not as flexible as older materials, so you never pressure test to pop-off like we used to. Never soak these new carbs in Gunk or any other strong cleaner. Don't even spray choke spray through them as it will ruin the accelerator pump o-ring and H side check valve. The only spray that is safe is a mild brake cleaner spray or electric motor spray. I don't know any other way to test the spray and I am not advocating this as a test, but if it gets in a cut on your finger and it doesn't burn it is OK to use in the carb!
So after tear down and inspection remove the metering screws and spray some of the brake cleaner spray through the H screw hole while holding the throttle open and you should see the spray coming out of the H nozzle sticking out into the venturi. Be sure to put on your safety glasse cause even brake spray burns your eyes.
If no spray comes out the H side is plugged and the main nozzle will need to be driven out and then the carb cleaned and then a new nozzle pressed in or better yet get a new carb because it is probably not worth trying to fix at this point. Now spray through the L metering screw hole and you should see the spray come out of the idle progression holes with the throttle butterfly held open. There are anywhere from two to four of these holes depending on the carb. This is how you verify if the H and L circuits are clear. The L side gets its fuel from the H side, which also feeds fuel to the accelerator pump. The little check valve in the main nozzle can be destroyed by harsh chemicals or by blowing through the circuits with a rubber tipped air gun at 100PSI.
Assembly of the HD on a 361 only has one gasket on the pump side. The diaphragm goes next to the smooth milled carb body so the two check valves can seal against the smooth surface. The gasket goes against the fuel pump cover so the bead on the cover can bite into the gasket. The opposite is how the metering side goes together: gasket to the carb body and diaphragm next to the cover, with the round disc facing towards the wet side or metering side of the carb. Inlet lever is set flush. Once the fuel pump side is together and the inlet needle and lever in pressure test to no higher than 10 PSI. Never do a pop off on current production carbs. Spray some of the brake cleaner on the metering chamber floor if the pressure does not hold to see if it is the needle.
The H check valve works by sealing closed at idle, due to high pressure in the venturi, so no air leaks into the metering chamber messing up the idle mixture. When the throttle is opened up air speed through the venturi lowers the pressure and the check valve opens up and fuel flows through the nozzle into the air stream. The only way to check the nozzle check valve and accelerator pump o-ring is to close both mixture screws, and pull a very slight vacuum at the bypass jet. I take the jet out and use tapered adapter to get down in the bore and pull negative 4 PSI and it should hold steady or not leak down to zero faster than a four count. If the H side is leaking it will not flood, it will screw up the idle mixture. The saw may act something like this: you warm it up, tune the WOT speed with the H screw, let it down to idle and set idle RPM and mess with the L screw for good idle and good throttle response, then rev to WOT again and when you let off it dies. Or it may come down to idle slowly. And you fiddle with the screws and you think it is good and then in two days it is all screwy again because a weather front came through and now the atmospheric pressure has changed slightly.
Just be sure the carb is clear, the diaphragms are not stiff or brittle, and if possible do the vacuum check. If the carb passes these tests then do a pressure/vacuum leak on the crankcase. I posted some pics of how to do this in another thread here I think.
To check the tank vent pull a vacuum on an empty tank and it should leak back to zero fast, but hold pressure. The vent is under a little secret window on the side of the tank housing.
As far as the ignition trouble you mentioned, the module has a micropressor in it and I have seen some weird failures from them. Just do a continuity test at the wires where they connect to the module and be sure the circuit is grounding. Also make sure you have the right resistor plug in the saw. If it is too hot and you have a lean fault that makes it hotter then it will run-on when the spark goes away.
Hope that helps...