I've found some probably unintended situations with the False Emperor events.
1. If you gain the HRE by a war for a claim as a non-Catholic, the Pope does not label you false emperor.
2. If you hold the HRE, are non-Catholic, and convert to a different non-Catholic religion, the false emperor event happens.
3. The pope offers a claim to the HRE to a non-Catholic Emperor of Francia.
(He even offers it to the false emperor he just denounced, if they hold both the HRE and Francia!)
4. If you have the false emperor modifier and dismantle the Papacy, you don't loose the modifier.
5. If an emperor who has the false emperor modifier looses the HRE title in a war (in my test, they were not annexed into the HRE), they do not loose the false emperor modifier.
6. If you destroy the HRE, you do not loose the false emperor modifier.
7. You don't seem to consistently loose the false emperor modifier if you convert back to Catholicism. Nor does the name change back.
8. If you convert to a religion, then back to Catholicism before the event fires, the Pope gives you all thee options. He shold probably still get mad, you converted religion, but you shouldn't be able to take the "spite the Pope option", and probably not the "Forced back to Catholicism" option either.

Also, my opinion is the modifier is too indiscriminate. Why should my Ásatrú vassals care what the head of a foreign religion thinks of me, and why does it gain me tyranny? I think a loss of legitimacy, gaining a relations debuff with all Catholics, and perhaps a loss of control in all Catholic provinces (if that can be done) would be more appropriate.