Try working backwards to eliminate what is working.
Clearly there is plenty of fuel getting into the system, hence the reason you are basically flooding the motor with fuel each time you pull the starter (that's why it's coming out the muffler etc).
First thing...is your fuel fresh??? Fuel that's "gone-off" is one of the greatest sources of non starting 2 stroke motors. Remember unleaded fuel loses it octane level (or ping) very quickly.
As you have put a whole new carburettor on the saw, you can pretty much eliminate it being a carburettor fault unless you haven't set the H &L screws to basic OEM settings.
As a side issue, whenever you play with the carburettor, always take a moment to check the old hoses closely as they tend to crack & perish with age. Simply remove them and bend them about to see if there are any cracks. In particular, check the impulse line as this line being cracked can often be the source of countless hours of frustration of a saw not starting or running correctly (and it's not always obvious).
Also while playing with the carburettor, always check the condition of the gaskets. While it's recommended to change the gaskets after putting on a new carburettor (or after a rebuild), it's not absolutely essential if the gaskets are not too old, or in very good condition.
My first thoughts would be check the electrical system. Start off with the spark plug, and then the coil, and lastly I'd check the air gap between the flywheel and coil. The reason I say this is because you are getting fuel into the system, but it appears there is no spark to make the fuel ignite. If it came into my workshop with the symptoms you describe those would be the first things I'd check.
Ok, so you've done this and the plug, coil, air gap setting are all fine. Next thing I'd check would be for air leaks. Low compression caused by a damaged oil seal, gasket or alike can also prevent a saw from starting and running. If the saw has had a lot of use, it can often be as simple as the rings being completely worn out.
Undertaking a vacuum test is always best to identify these faults, however most guys doing DIY home repairs don't have the right specialist tools to undertake this task. In that situation just simply use an automotive compression test gauge. While not as precise, it will identify if you have an air leak in the system or low compression.
Try these simple things first and let us know how you get on.
Also, just make sure the "fuel tank vent" is in place and not blocked. It's a simple thing, but I've seen a few of the old larger chainsaws (mainly Husky's and Jonsered's) having tank vent issues.
