Which is the least massive object?
Quoting Wikipedia,
In the hierarchical, restricted three-body problem, it is assumed that the satellite has negligible mass compared with the other two bodies (the "primary" and the "perturber"), . . .
This is the case studied in Kozai (1962), specifically, the case of asteroids being perturbed by Jupiter. While not massless, the difference in mass is large enough that the asteroid's mass is negligible.
How does the three-body system evolve? . . . Whose inclination and whose eccentricity?
Wikipedia is again rather direct in this, stating that the conserved quantity $L_z$ depends on the satellite's orbit's eccentricity and inclination:
$$L_z=sqrt{1-e^2}cos i$$
As the satellite's mass is negligible, it will not have any significant effect on its perturber.
Can it become circular, eccentric, circular, eccentric?
This is basically asking if the eccentricity (and therefore inclination) can be described by some periodic function. Again, this is given in the Wikipedia article. A slightly different but just as simple xpresison is given in Takeda & Rasio (2005):
$$text{Kozai Period}=P_{text{perturbed}}left(frac{m_{text{star}}+m_{text{perturbed}}}{m_{text{perturber}}}right)left(frac{a_{text{perturber}}}{a_{text{perturbed}}}right)^3(1-e_{text{perturber}}^2)^{3/2}$$
In the approximation discussed above, in cases of extreme mass difference, $m_{text{perturbed}}to0$.
The inner binary will lose energy during the whole process, right?
The whole thing is periodic, so no energy is lost.
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