How do I kill the robot by the third door in the Institute Ruins bunker?
I try to dodge in, throw a grenade, and dodge out, but it chases after me and kills me.
Even a grenade barely makes a dent.

I'm stumped and can't get any further in the game !!

This worked for me: I put all companions further away in corridor (told them to wait for me so they wouldn't interfere), I loaded Vintorez rifle with AP ammo and once fight began I fired burst on robot, retreated behind corner so robot would have to chase me around corner and waste it's Action Points on movement and not firing and then I fired burst again which destroyed robot like charm. I tried it approximately three times to test it and it always worked.

Vintorez rifle (scoped variant) can be obtained in encounter in Roaring forest (a little difficult since that guy wielding it can pretty much one shot you with it's burst fire but the gun is worth the trouble ;) ). In case you missed it vendors and occasional caravan traders tend to have assault variant of Vintorez (non-scope) but it is also very rare, I remember seeing it like once or twice maybe.

I had saved the experimental VSS rifle and this was the perfect opportunity to use it on, thanks for the advice. Put my companions back, took one step in, first round burst critical'd him down to 5 hp, step back into corridor, switch back to SKS, dead.

Thank you!