yes there are fog lights installed on the touring model... i think this is a factory option Here are the instructions for installing the fog lights; http://www.handaaccessories.com/odyssey/05odyfoglight.pdf
bad cluster
Defective or misaligned brake light switch.
To dim the dash lights on a 2004 Honda Element, locate the dashboard light dimmer switch, which is typically found on the left side of the steering wheel, near the headlight controls. Turn the dimmer switch clockwise to brighten the lights or counterclockwise to dim them. Adjust it to your desired level of brightness. If your Element has an automatic feature, the lights may also adjust based on ambient light conditions.
No you do not need a battery, but the lights and horn will not work unless you have a battery installed, dead or not.
find the burnt out fuse and replace accordingly with the same or weaker: NEVER stronger.
Honda Element was created in 2003.
In a 2003 Honda Element, the fuse for the tail lights and dash lights is located in the fuse box, which is found under the dashboard on the driver's side. Specifically, you should check the fuse labeled "TAIL" or "DASH" in the fuse diagram on the cover of the fuse box. If you need to replace the fuse, make sure to use one with the same amperage rating as specified in the owner's manual.
The warning lights on a Honda Accord are the same as the warning lights on most other cars.
check to see if you have power going to all the signals with a tester
how do i wire my brake lights on my Honda 1000
http://www.autotalk.com/how-to/honda-element/honda-element-03-04-keyless-remote-programming-instructions.html