My 2002 dodge Durango the abs light , the brake light, and the air bag lights stay on making my air conditioner and speedometer not work.

Most likely the Rear axle speed sensor in the diff area

The brake and abs light come on if a problem is detected. When the abs light is on the abs is NOT working. You need to pull the codes from the ecm to detect the problem

wheel bearings are wore out so abs isnt working. change them and the abs light and spedo will be fixed. peddle is low because the fluid goes through the not working abs box.

Number one cause is low brake fluid check and make sure brake fluid is above the minimum line if not fill it to past the minimum line with whatever the DOT specified brake fluid on the cap says

A CLOCKSPRING FOR THE AIR BAG COME

check for codes with scanner it most likely is the abs sensor on the rear differential housing that is bad easy fix

wheel speed sensor data may not be getting to t e abs control module.
The hand brake light may stay on if the hand brake is not fully retracted or the cable is loose.

Check brake fluid level to make sure you have enough in the master cylinder.

NEED TO HAVE SOME ONE CHECK THE CODES,SOUNDS LIKE YOU HAVE TWO SYSTEMS WITH MALFUNCTION,AIR BAG AND ABS,SO IT MAY BE A CONNECTION PROBLEM

The ABS stands for (Anti-Lock Brake System). The air bag light or SRS stands for (Supplemental Restraint System). The lights are staying on because there is a problem with both systems. They have nothing in common with each other. I suggest you have this looked at by a professional. Working on the SRS can be hazardous so leave that to a pro.

If you backup lights are not working as well, I would say check the fuse for the backup lights, if the fuse is good, see if you are getting power to that fuse. We had this problem and after weeks of searching and changing sensors, we found the backup light fuse was good, but not getting power. Ran a wire from that fuse to the cig. lighter fuse and all woprksf great.