Is this normal? What about the sterile cockpit rule? Or is that rule only on paper and in reality it isn't followed by the majority?
Chit-chat is fairly normal. It's more of a concept except that in some very "happy" companies, it's a hard rule. In many companies it's not enforced with much vigor.
But it's reasonable to have a discussion on the the value of "sterile cockpits". Some believe that aircraft are so difficult and complicated to fly that safety can only be ensured by nothing less than total concentration, all of the time. I'm sure that these are the same people who also believe that nothing other than documents relevant the flight are read in the flight deck during flight.
The extreme opposite are those chat the whole time, watch videos, listen to music, read comics and newspapers and play silly games.
Now you would have thought that there would have been a difference between the safety record of the "professional" crews and that of the "gash amateurs". But I can't find one. I also fail to find any differences between either types of crew in their reaction to unexpected events. Both appear to handle the "startle factor" equally well (or poorly).
Personally, I believe you have to have an appropriate level of arousal at the appropriate time. Total concentration is a poor as none and during each phase of flight a different level is required. Therefore, I'll suggest that a very non-precise "enough" level of concentration is required. And some crews are able chat at times when others have to shut up because they can't fly and talk.
PM