I think the problem is in the iPhone's logic board headphone detection circuitry. There might be a short on the board making it look like there is a set of headphones always plugged in. You would need a new logic board to fix this problem.
You probably cut or ripped your display (LCD) panel cable. To remedy this problem you would have to get a new display (iFixit Part No. IF137-001-1). You can make sure that the device actually works by turning the vibrator switch on and off and if you feel any vibration, then the display is the problem, not the phone itself.
When your phone got into the water the battery contacts shorted out and destroyed the battery. You can get a new battery from iFixit and install it yourself. This will usually correct the problem, but there is a slight chance that the logic board is damaged. I would first get the battery, and if that does not work then you would have to get another logic board for your iPhone.
For the SIM card issue, This seems like a SIM card problem. If you have another SIM card try it in the phone. If this still does not work the SIM card holder might be bad, or the logic board might be bad too. For the battery, it probably has lost it's ability to be charged. If you get the SIM card problem fixed you can replace the battery and that problem should be solved too.