> Isn't there any API (? or something) for apps to run in "minimal" mode, to wake up the phone when needed?
Ubuntu Touch is the only distro that has a mechanism like this, and my understanding is it doesn't run that well on the PinePhone. The rest will wake up for an incoming call or SMS, but otherwise will stay asleep.
> Does this also mean that it's not possible to have the phone notify you when you receive an email in sleep mode?
Currently, AFAIK, yes.