These are excellent questions. My thoughts are:
"* Should the functionality be configurable by the user (allow/deny)?"
Yes. There is no duty to contact the emergency services, nor to give any information you don't want to. If a user wants to use the "basic" emergency call system, then they should be allowed to, at their own risk.
"* The user may be unaware that extra stuff was activated. Should it be switched off when the call ends (if it was off)? X minutes after the call ends? A notification presented to the user with a choice of "keep on / switch off"?"
I think that would have to mandated in the specs for all sorts of reasons. I think a combination of switching off after X minutes and a user-operable choice is optimal.
"* Should battery level be checked so that activating extra stuff won't drain it too fast?"
*Any* call is better than none, and so the phone signal should be prioritised, then wifi, and GPS only if there is sufficient power after the other two have been switched on.
"* Is it so difficult to imagine cases where the caller - or the owner of the phone used - might want to place an emergency call but remain anonymous and unlocatable? The emergency may not involve him/her directly, the services may not need to be deployed to precisely the caller's location, etc."
This is a common situation (consider an elderly relative ringing up to say they don't feel well, and then going quiet mid-sentence), and so there needs to be an override to a non-automated system.
"* If there is a mechanism to activate precise location beacon when dialling a specific set of numbers, who will convince me that it cannot be done by, say, making a call or sending an SMS to me? Or, say, by pressing a particular password/PIN (joel's_123_backdoor or something?) on the locked screen?"
Yes, it will be possible, but I would say this is going to be somewhat self-limiting. The battery isn't going to last any time if the whole range of locating devices is switched on (my Galaxy Note will do about an hour with the GPS switched on). Even the most clueless of users is going to notice their device repeatedly becoming dead after a short time (though what they would do about it is another question!)