Chainsaw Repair

That rubber bellows is the intake and is very common in chainsaw design.   It helps with any flexing that occurs with use. 

I would get the piston and crankshaft installed into the cylinder half first then use a sealer like yamabond 4 or some other sealer that is fuel proof and then put the two halves together.  It does not take alot of sealer just a nice even thin coat.    Have all your screws ready to go to tighen down the two halves.   Also after assembly let it sit for about 24 hours to let the sealer cure.   Then you can finish re-assembly and fire it up. 
I always spread a thin coat of 2 stroke oil on the cylinder walls when putting the piston back in the cylinder.

Yikes. I'd have declined this job also as the 440 isn't enough saw to warrant this much effort. And I'm a full time shop. How many hours do you have into this job?

Completely filling an oil tank really isn't the proper way to test for a leak. We fill the tank 2/3 with oil and let it sit overnight. If the tank is topped off, then the vent is covered and that will likely cause the saw to "push" oil. Big temp swings make this worse as the oil expands. Stored in a truck toolbox, shed or any place that will have a big jump in temp when the sun hits it almost guarantees a leak on a full saw. You might go through all of this and still have the same problem.

Did you quote a price for this work to the customer first? If so, then you must have quoted incorrectly, or did they customer just say 'get on and fix it' with no mention of cost?

This is a very labour intensive job, and I would have been looking at it thinking is this a mornings work, or a full days works or somewhere inbetween? A mornings work for me is 4 hours, so four hours labour plus parts minimum, should be enough for the customer think 'this isn't worth doing unless I do it myself' I would rather make money than work for nothing, so jobs like yours, that can also have hidden additional issues that may require more parts and labour I generally over quote to cover myself- if the customer says go ahead, then I am covered if he doesn't then I haven't lost anything as it would have cost me money to do it for less just to get the job.

Sometimes you have to say no, and be honest with the customer.