2005 dodge neon sxt dashboard light came on and won't turn off. I dont have a car manual but the light indicator looks like a radiator. I added coolant and the light still did not go off.

P0010 code is the camshaft position sensor. If this fails the vehicle will simply shut off.

P0315 - Crankshaft Position System Variation Not Learned
P0335 - Magnetic Crankshaft Position Sensor Electrical Error China OBD2 car diagnostic tools wholesale on CareUCar

Does it only come out of the vents when the A/C or heat is on? Or is it constant no matter what?

Ok, even if the battery is new, get a voltmeter and measure the battery, if its below 12v, thats the problem. That way you completely eliminate all these factors without a doubt. Also check negative wire from battery to the ground box and connections. It sounds more like a corroded terminal.

change fuel filter & air filter

sounds to me like it has jumped timing

The red circle means there is an intermittent problem with your vehicle security.If your car starts and runs fine it would be hard to diagnose until it wont start at all.

if you are the original owner, look in your paperwork. if you cant get to the original owner or they dont have the info, you can go to your dealer, and they can reset it for you. and charge you

or you can just start entering codes till it works, which will take you weeks or more.

good luck

Double check the fuse. If it is good. then you got a bigger problem. Like a short in the wire or your instrument cluster.