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no, this has nothing to do with the gearbox.
the oil light means that oil pressure is low or there is a problem with the oil pressure sensor.
the battery light means that the alternator is not charging because it is faulty, the drive belt is off, or the battery is bad.
the eml light means that there is a problem with electronic throttle control, although this may or may not be related to the alternator issue.
the engine cutting out can be for many reasons - failing fuel pump, failing engine sensor, etc.
again, this is nothing to do with the gearbox. I would strongly suggest you take the vehicle to another garage for proper diagnosis and repair.
Please let me know if you have further questions. Please do not forget to click a positive rating as this is the only way I receive credit for my answer, thank you. Best wishes with your vehicle.