Could be. But it might be an alternator problem instead of a batery problem. You need a good alternator to keep the bttery recharged. I wouldn't run out and buy a battery without getting the alternator checked.
If the cigarette lighter would normally work without turning on the ignition, yes.
So it would seem. If the ignition is in the column it may also be jammed by the steering wheel lock. Try turning the steering wheel and holding it while turning the ignition key.
A number of things can cause hesitation. There could be water or dirt in the tank. Also, it could be an ignition problem. Your ignition timing may be slightly off and your plugs are firing a split second off top dead center. This would account for the hesitation and subsequent bucking.
It could be a bad battery.
A malfunctioning the ignition switch will allow the car to keep running after the key is turned off. You can remove the positive ground cable from the battery to shut the car off.
Could be your battery, starter, ignition, check them... would start with the battery !
Headlights run off the battery not the ignition...
The voltage reg is sticking and will run down the battery.
Your ignition relay might be bad if you have good power at the battery, but nothing happens at the starter when you turn the key, all the lights light up and everything appears normal, but the starter will not engage. You may also hear a click when turning the key to start, but nothing else.
Use of the side and rear hatch doors have a dramatic drain on the battery. We have found that whenever we go on trips, or camping, where the doors are frequently opened and closed, the car the battery will drain. So far, our only solution is to disconnect the battery when the car will not be driven or charged between frequent door use.
Dead cell in the battery. A light is on drawing power from the battery. You may also have a relay that is stuck.
You would need to rewire them. They should be on with the ignition in the accessory position.