Re: stator
The charging coils of the stator are heavy windings and usually relatively trouble-free. It will usually either be good or bad, charge or no charge, no in-between. before you look to replace it check out all other parts of the system, including the voltmeter. You may have a high resistance connection giving false readings or an inaccurate gauge.
Check the battery after running and sitting a while--if it reads 12.5 to 13 volts or so, then it is being charged properly. If it reads less than 12, either it is bad or indeed it is not being charged correctly. Then, put it on a battery charger. If it now measures over 12 volts upon sitting, it is good and not being charged by the engine correctly.
And to answer the first part of your question: Yes, the stator will swap onto your engine.