This is just a partial answer (so far), but it will address at least part of your question.
1.To what temperatures the dark surface of such planet could be heated? Is there possibility of liquid water?
Wikipedia gives a good estimate for this: 1,200 degrees Celsius. For a planet, that's pretty hot! Any water there would evaporate very quickly. In fact, it is expected that a lot of the planet's surface is molten. Now, does this apply to the back side of the planet? I would think so. While only the front side of the planet would bask in the warmth of Alpha Centauri B, the heat should dissipate throughout the entire planet.
2.Will the radiation of the second star be enough to provide normal day-like illumination and heating?
The short answer here is no. In fact, when compared to our solar system, nothing about the radiation Alpha Centauri Bb gets is normal. It is situated extremely close to Alpha Centauri B - 0.04 AU. Given that both Alpha Centauri A and Alpha Centauri B are similar to the Sun, I think it's clear that this exoplanet will not receive a "normal" (normal here meaning similar to Earth) amount of radiation from Alpha Centauri B. Conversely, Alpha Centauri Bb is 11 AU away from Alpha Centauri A - much greater than the distance between it and Alpha Centauri B (a bit more than the distance between Saturn and the Sun). This means that the planet will receive only a tiny fraction of light from Alpha Centauri A relative to Alpha Centauri B.
4.Will such planet experience seasons and how they would be arranged?
Seasons depend on the tilt of a planet's axis. Unfortunately, it is hard (if not impossible) to measure the tilt of the axis of Alpha Centauri Bb (I haven't been able to find any measurements). I would assume that it would experience some seasons, but I don't know how much they would extend.
As for question 3 - I'm not quite sure what you mean by "calendar." If you mean an artificial calendar, I would think that any beings on the planet (life there is extremely unlikely) would only use Alpha Centauri B to determine their years (just over three days and five hours).