I have a cigarette lighter hole in my car that produces 12V. I bought a cigarette lighter extender that has 3 cigarette holes and each hole has 12V so that's 36V instead of 12V. Will that damage my battery? What are the cons to my battery or engine of having that?
When you purchase a lighter plug extension that adds three extra plugs, you're not changing the voltage because the plugs are in parallel with each other so that each plug can deliver 12V to the accessory connected. How much current each accessory draws, however will be added together, and can't exceed the current capacity, usually dictated by the fuse.
Let's say the lighter jack in your dash is protected by a 15A (amp) fuse. You plug in your extension and plug three accessories into the plug. You connect your cell phone charger to one plug. If you have a recent phone, it probably draws about 2A at 5V (your adapter reduces the voltage from 12V to 5V) which is just under 1A at 12 volts. You connect a GPS to the second plug and it draws probably half an amp. In the last plug, you connect an electric cooler which draws about 7.5A. In total, you're drawing (1A + 0.5A + 7.5A) 9A of current, so you're good.
If, on the other hand, you try to power three coolers, each pulling 7.5A, you're going to blow your fuse.
I have a cigarette lighter hole in my car that produces 12W
This is not true. The cigarette lighter port/power outlet outputs 12 V (volts, not watts). The power consumed will be determined by the appliance that is plugged in.
The number of power outlets in the vehicle is of no consequence; if nothing is plugged into them then they will have no impact on the vehicle's battery.
The socket on the car will be protected by a fuse, if you draw too much current you will blow the fuse, so there should be nothing to worry about with the car. However if it is a cheap adapter and you draw a lot of current through it, the adapter may start to get hot. If you are powering electronic items like a sat-nav and not 3 cigarette lighter elements, then you are unlikely to overload the adapter.