I think you must begin by accepting that you do not have a Lawnmower Problem but actually have three separate problems...
The Battery Charging system is obvoiusly not working.
There appears to be an issue with the circuitry that kills the engine
The engine runs erratically with varying speed indepentdent of any operator input - possibly fuel related.
It is often best when faced with multiple problems to address the one that is most likely to be the easiest to resolve - early success raises the spirits and gives a positive expectation for the next problem.
Good luck...
Peccavi
July 2009