Chrystler Sebring JXi Convertible Battery Problems

My car is a '99 Chrystler Sebring JXi Convertible with about 117,200 miles on it. My father bought a new battery for it about a year ago, but recently I have gone to my car to start it and found that it was dead and heard a "clicking" noise when I tried to start it. This has happened about 6 or 7 times. It is usually accompanied by my odometer flashing zero while it is starting, and it starts slower than usual. I don't think it is the alternator because once it is jump started, it works fine and doesn't die. Also, when I turn the car off, a sound comes from the hood that resembles a "gurgling" sound, and I opened my hood today and saw a little bit of white smoke and the gurgling noise coming from around where my coolant tank is(passenger side). Also, I have been using a radar detector in my car lighter and all was well, until I accidentally broke the end of inside of the lighter. In an effort to remove the piece from the inside I pulled out a cord (it was meant to be removed) thinking I could remove the whole console itself but that didn't work. So I have left that unplugged and have not used my Car Lighter, Do you think that this is eating up my batterty in anyway? Even though it is unplugged? Thankyou,
Zach Edmondson.