Safely powering a relay from same source as attiny/arduino?
do I have to duplicate that circuit just for the relay?


No, not really, no.


  Is the ATTINY capable of triggering a 12v relay via ground?


The ATTtiny is capable of triggering a transistor - the transistor is capable of triggering a 12V relay.

Unless your relay is incredibly high powered and the coil takes (and generates through back-EMF) massive amounts of current, there really is no problem running it alongside a small microcontroller. Just be sure you have the flyback diode, and maybe a damping capacitor as well to absorb noise.

TBH you're probably going to be getting more noise from contact spark than you are from the relay's coil.