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If you send it to another calculator, it will not expire. What I'm calling the expiration count is a field on the certificate page; it's not stored within the app, but it's set by the OS at the time the app is installed. For normal, non-limited-trial apps, it should be set to the value 80 00. Since Axe circumvents the normal installation procedure, this doesn't happen, so the field is left at its default value (FF FF), which is interpreted to mean that the app should expire after being run 16 times.
Well, as "easily" as writing to other areas in Flash. (And yes, that's kind of dangerous; if you screw up the certificate, it's possible to crash the boot code and make it impossible to install an OS.)
I'm not entirely sure, actually. BrandonW devised a technique that you can use to run arbitrary code by sending specially crafted link packets. But in order to actually fix the damage and reinstall an OS, you would then need to unlock Flash somehow; this can always be done on an 83+ SE or 84+, but I don't know if there's any way to do it on an 83+ BE or 73 with no OS installed.