If Method2 takes longer that the Page life cycle to complete, will my Page life cycle complete and send its html output to the browser?
Yes.
You'll need to ensure that the handler does not finish execution before you are ready for ASP to move on and continue with the page lifecycle process.
You should also avoid the manipulation of shared memory between threads, as you are not doing so in a safe manor.
Rather than setting a success
boolean, you can simply cut out all of that code. You can let the exception bubble up and have the main thread check to see that neither task threw an exception instead. When doing that, the TPL will ensure proper synchronization of the data shared between threads.