Your proposed situation is not plausible. Gravity extends forever into space. There is no distance beyond which stars do not feel each others' pull. If a planet can travel back and forth between the stars repeatedly, then the stars must be orbiting each other. There is no difference (except in strength) between the gravity of a star and that of a planet.
Such a planetary orbit probably would not be stable for long, though some truly bizarre orbits are possible, especially if there are not any other planets of any size--they, too, would contribute gravity affecting both stars and the planet in question.
The size of the planet is not a major issue, especially if there are no other planets around. But in systems where planets have highly noncircular orbits, small planets tend to be ejected from the system by large ones. (Here, you could view one of the stars as a "large planet.")
As for life, who knows? Probably any planet with reasonably sustained earthlike temperatures and a protective earthlike atmosphere could harbor some form of earthlike microbe. But we don't know anything worth knowing about life on other worlds yet.