Phone will not turn on

this generally indicates a bad battery, try to purchase a replacement battery
if this does not fix the problem, then sorry to say, but the phone is shot, it is older now

You can fix that by flashing new software into the phone, try taking to a Verizon Wireless service center and they will either flash software or replace unit.

Car kit mode means one of two things, first it could have a piece of lint or paper stuck in the charge port making it think there is a car kit plugged in, or second it could be that some moisture got into the port and its giving a indication that something is plugged in. and thats why you're not getting any sound. There is no specific setting to turn on or off 'car kit' mode you can use a paper clip or something and mess with the port a little bit to see if 'car kit' goes away or replace phone

You'll need to send the phone to Cell Phone CPR to have it properly repaired.
Here's the link:
http://www.cellphonerepair.net/

my guess is mate that your fone is dying

reload the software

You are going to have to contact verizon wireless and hopefully you are still under the manufacturers warranty. You might be able to reflash it, but that requires purchasing a data cable and software.

it's possible your battery is mot putting out enough to run your phone. if you didn't drop it or get it wet.
Try replacing the battery and see if that handles your problem, I suspect it will

This is an obvious software issue.
I took mine down to Verizon to ask them to reboot the software.
The tech said that unlike a computer, there is no BIOS chip.
So they have to pull the motherboard to reboot.
And that was not within his ability.
So he pulled a field replacement from his cupboard.
Put my Battery and backplate on it.
And assigned my phone number too the new phone.
All at no charge.

Yes. You have to contact VERIZON or another cell phone dealer and ask them for the number to LG or what ever company your phone is made by. In my case it was LG and they sold me a replacement cell phone camera lens, with special glue, for around ten bucks. It was mailed directly to my house.