In my Company, as in most I believe, we fly leg and leg about. The selection of sector is as a matter of courtesy usually offered to the FO subject to any particular judgement of the skipper on the day related to currency, weather etc.
PF briefs the departure - if this is the FO, I would expect him to retain control of the aircraft in any emergency and follow his briefed intentions with whatever support and co-operation I can provide. Once the immediate 'crisis' is over and in all probability the autopilot is engaged, I MAY elect to fly the aircraft as is my perogative as Commander. If it is a straightforward problem I would invariably be happy to let the FO continue as PF - if we are double asymmetric or things are really bleak then I may take control - it all depends on the day.
All self-respecting occupants of the rhs want to achieve commands - they should be given every opportunity to develop their skills of all kinds to be ready for the sudden phone call from the Chief Pilot that says 'your command course starts tomorrrow' - yes, I know that in this day and age, such calls are few and far between !!!
Sometimes F/O's may feel they are going through the motions, sometimes they feel less than ready to develop their role - I flew with a chap not long ago who was horrified when I asked him to
give a 'welcome PA' to the passengers - he almost started shaking - what he would have done if an engine failed at V1 - well I do wonder !