First up, changing from 5s to 6s is as simple as plugging in the 6s battery. In BECS you generally do not need to change the input voltage, just the output voltage you require thus a change from 5s to 6s is as simple as plugging in the 6s battery. Just check beforehand what the spec of your equipment is ie if you motors which get a direct feed are specced at 5s only then you will need to change these. Another thing to bear in mind is that by moving to 6s the motor rpm will increase.
As for batteries, I think that using voltage to judge the amount of flight time left is completely the wrong way to do it. The best way to do it is by judging how much you need to put back into the battery. The golden rule with lipos if you wish to maximise life of the battery is the 80% rule. So, if you have a 10,000mAh battery you should not be putting anything more than 8,000mAhs in when re-charging. To work out your flight times, conduct let's say a 5 min flight, recharge and note how many mAhs went back in, divide by 5 to get your mAh per minute and then divide that into the capacity of the battery. Simples. This is what you then set the timer to on your TX. The other advantage to this is that you know that when the timer goes off on your TX you still have sufficient power left to immediately return to base. For those guys running down to 3.2 v per cell, I think this is quite dangerous as you are now getting into the zone where ESCs may start to automatically shut down (be it sudden or a gradual reduction in power). One thing is for sure, you are shortening your battery's life when going to 3.2v per cell.