How dangerous would it be to take off at night with no ATC, no runway lighting system and have a single engine working?
I was wondering how dangerous it would be


When you preface a question with the above, you're asking for opinion, and when it comes to the details of what you ask, there will be widely differing opinions. That said, and in the context of the U.S., and if I understand your question (apologies if I'm misreading), the details of your question break down into three separate areas. One can come up with a number of possible scenarios which might combine those details, but let's use a concrete instance to better explain. I at one time was the company pilot for a firm that manufactured and maintained computerized sawmill equipment at mills throughout the west coast of the U.S. If one of our computerized systems broke down, we would fly to the mill as soon as possible. Some of these mills had private airfields alongside the mills or nearby. We could not reach ATC on the ground from any of them, and most had no runway lighting. Typically we would go into an airfield during the day, and often it would be dark by the time we completed repairs and left.


  no ATC


All night flights were done IFR. We would get an IFR clearance over the phone with a "void if not off by" time. So you takeoff, abiding by the clearance, and contact ATC on the assigned frequency as soon as you have enough altitude to reach them.


  take off at night when you have no runway lights


As long as you have landing lights and are familiar with an airfield, the lack of taxiway and runway lights is not a significant problem. What is a problem at remote runways is the possibility of animals (usually deer) on the runway, which is why we typically had at a car precede us down the runway while flashing their lights and honking.


  an engine failure (with a twin-engine) during take off at night


The twin engine aircraft we had available were a Cessna 310 and later a King Air A90. Unlike air carrier airplanes, continuing the takeoff if an engine failed during the takeoff roll was not an option, but the probability of that happening wasn't affected by time of day. Day or night, the only option was to abort. We were no more exposed during the night than during the day in that respect.

If an engine failure happened after takeoff, day or night, you would handle it in the same way. Fortunately, we were operating out of airfields where our departure path wouldn't need to be altered due to the reduced single-engine climb rate.

We had a single-engine Cessna Turbo 210 available as well. During the time that was our primary aircraft, we had no qualms about single-engine night or IFR operations. Many may disagree with such. However, although as I remember Part 135 of the FAA regs does not allow single-engine night or IFR, Part 91 does (or at least did) and I never personally knew anyone who had problems while doing so. Obviously there is some additional risk, but we considered that acceptable.