: Are they, themselves, alive? Were they ever?

: Have we been told that they created life? As opposed to tinkering with life.
: Did they create all life that has ever existed? Or just some life?

No one really knows. The idea floated by the Forerunners is that the Precursors "accelerated" the evolution of sentient life, but considering that they also tell us the Precursors peaceably left while the last Precursors say the Forerunners killed them off, it's impossible to say with any certainty what happened besides the Flood apparently being the Precursor's doing.

It's fair to say that since they reduced themselves to a powder form that eventually gave rise to the Flood, their frame of reference for "living" might be different from our own. The Domain itself could be a form of life to them (it's treated as such by the Forerunners).

: Is Cortana alive? What is life?

Frankie has some opinions about that in Saint's Testimony :)