I used to ride enduro in the 1980s on a KLX 250 :>)
Sounds like you have two issues.
Your spluttering and jolting sounds like a defective Reed Valve, but could also be carburetor float level problems. If it is runs fine when on the power band, then the carb is fine. For the cost, I would replace the reed valve. http://www.athenaparts.com/eng/sections/view/128/carbon-fiber-reeds-for-intake-reed-valve/page:6
Regarding the electrics, this could be a few things. Check all the earthing points, especially the one on the negative side of the battery. Take the wire off where it grounds to the frame, clean and apply Vaseline which stops corrosion and is electrically conductive. Check all the connectors that you can find. Pull them and make sure they are clean and sealed. Vaseline is good on these too. The specs below confirms it is rectified and regulated. Check the output voltage of the alternator with a test meter. Check the battery is as spec Battery 12v 3Ah and get it stress tested, or just replace it, if it older than 3 or 4 years.
You really need a good workshop manual. I had them for all my scooters and bikes.
Spare Parts List http://www.cmsnl.com/kawasaki-kdx125-a3-kdx125sr-1992-europe-nr_model11759/partslist/
Voltage Main Fuse 20A
Ignition CDI electronic advance
Charging System 3 phase AC, 27V@4000rpm. Rectified 14.5@4000rpm
Spark Plugs NGK BR8ES
Bulb-Headlight Battery 12v 3Ah
Ignition Timing 10.5BTDC@1300rpm - 26.5BTDC@3000rpm
Engine Type Liquid Cooled 2 Stroke Reed Valve Straight Single Bore x Stroke 56 x 50.6 mm Compression Ratio
Lubrication System Oil injection
Carburation Main Jet 115
Carburation Idle Jet 48
Pilot Screw Setting 1.5
Red line RPM 10,500rpm kawasaki KDX 125 Google Search