Yes there is a 'plug-in' heater that can be used ( think it was made by Salamander). During summer my boiler is switched off completely, so no gas bill for hot water.
Yes you are right about the bag! The toilet will not flush properly if you put the bag in the cistern. There will not be enough water to do the job properly.
This brings back memories!! Yes there is a divertor valve in the boiler and yes if it is sticking it will work ok in heating mode but no hot water. All combi boilers, unless they have a twin primary heat exchanger (Worcester i Junior) default to heating .