Date: 201306
Local time of day: 1801-2400
Location: ZZZZ.Airport, FO
Flight conditions: IMC
Weather elements / visibility: Turbulence
Light: Night
Aircraft: B757-200
Operator: Air Carrier
Operating under: Part 121
Flight plan: IFR
Mission: Cargo / Freight / Delivery
Flight phase: Cruise
ATC / advisory: Center ZZZZ
Reporter organization: Air Carrier
Reporter function: Pilot Flying; Captain
Reporter qualification: Flight Crew Air Transport Pilot (ATP)
Second reporter function: Pilot Not Flying; First Officer
Anomaly: Aircraft Equipment Problem Less Severe; Deviation - Altitude Excursion From Assigned Altitude; Flight Deck / Cabin / Aircraft Event Smoke / Fire / Fumes / Odor; Inflight Event / Encounter Weather / Turbulence
Detector: Person Flight Crew
When detected: In-flight
Result: Air Traffic Control Provided Assistance; Flight Crew Regained Aircraft Control
Contributing factors: Weather
Primary problem: Weather

Narrative: In cruise at FL350; embedded in 'milk bowl' conditions radar painting solid green and in moisture. Within a couple sweeps of the radar we were embedded in solid red and were in Heavy Mod to Severe Turbulence and Lightning. Experienced excursions of +300 to -500 feet in altitude while trying to override manually. Immediately called ATC (the First Officer) and descended to [FL]330 and turned shortest direction to green radar returns - no improvement. Two concurrent events - strong electrical smell (as in burning) and LEMER Door EICAS along with overhead light - EMERG DOORS. The odor cleared immediately and we found no possible causes in the cockpit. We also began monitoring cabin altitude. ATC approved the descent and indicated there was no traffic for 'miles;' and a turn south toward ZZZZ to clear the weather and pre-emptively (to me) in event of further problems. No emergency was declared. On the descent we received a Master Caution of GEAR NOT DOWN at 6;000 FT all the way to gear extension. Also; Captain lost RadALt and First Officer lost ASA. On landing were informed of engine exceedances in flight at FL350 - numbers unknown but; no indications in the cockpit. Wrote up all the pertinent points and debriefed Maintenance who grounded the aircraft for detailed inspection to include lightning strike and overstress.This report is submitted to document the altitude excursions due to severe turbulence. I believe all the related maintenance issues are tied to the turbulence and a possible lightning strike. First Officer and I conducted a thorough debrief both in flight to double check each other and on the ground to review what we could have done. A totally unexpected and unpredicted weather related event is my end assessment. There appeared to be no issues with ZZZZ ATC and with no related or continuous concerns after the odor cleared I elected not to declare an emergency and landed uneventfully.


Second narrative: [No additional information contained in this report].


Synopsis: The crew of a Boeing 757-200 unexpectedly encountered moderate to severe turbulence and lightning while in cruise at FL350; resulting in; among other things; altitude deviations. A strong electrical burning smell permeated the flight deck; and several random emergency and EICAS lights illuminated; suggesting a possible lightning strike.