I recently relocated to Israel for post-doctorate research, on an A/2 Student VISA. Am I eligible for a permanent or temporary Mispar Zehut (national ID number)? If not, given that most services require this, is there an alternative I can use (e.g. passport number)?
To be honest I'm having a hard time finding information about this. As far as I can tell the "Population and Immigration Authority" is responsible for ID-cards.
I'd advise you to get in touch with them on *3450 or 1-222-3450 (from an Israeli phone line) or on +972-2-6294666 (from a foreign number). The call centre is be available for Hebrew, English, Arabic and Russian speakers. Calling hours are Sunday to Thursday, from 8am to 4pm local time. So for today you're already too late to call them and tomorrow they likely won't be working due to it being the eve of the last day of Pesach.
I know from familial experience that it is possible to get an ID-card as a foreigner however that was over 30 years ago, so I'm not sure what the current state of affairs are but seeing as it's still valid I'd assume it should still be possible.
If however it isn't possible you can just say that you don't have an Israeli ID, usually whomever needs it should have an alternative for you. Either your foreign ID number or maybe your phone number might be a good alternative depending on what they'll accept.