Does the LA County or LA City Fire Department still use Open Cab Fire Engines as seen on Emergency?

Most likely, no (I'm not counting parade pieces)

NFPA has mandated since the early 90s (maybe earlier) that all new fire apparatus manufactured after that date provide for fully enclosed seating for firefighters. I can't recall seeing any fire truck manufactured after about 1990 that even has open jump seats. Completely open cabs went the way of the dodo back in the 70s (in terms of manufacture--some have remained in service since).

Since most large departments only keep apparatus in front line service for 15-20 years (sometimes less), and fully-enclosed cabs have been required for at least 20 years, I think it is very unlikely that they still have any trucks on the road with open jumpseats.

Besides exposing firefighters and equipment to the elements, open cabs don't provide any protection in the event of a roll-over.

Since the riots were 20 years ago, you weren't seeing things. Open jumpseats were still common and I recall seeing So Cal fire departments riding around at the time with a few open cab engines with some sort of jerry-rigged roof on them.

You can run through pictures on their websites and try to find an open cab if you want:

http://fire.lacounty.gov/HometownFireStations/Home...

I'm sure they do. Fire trucks don't run up a lot of miles over the years and it's quite common for a fire company to be driving a rig that's 20-30 years old, and still running like new. Unless the truck gets in a wreck, or the company that built them doesn't make parts for it any more, there's no real need to retire or replace them.

Since Los Angeles doesn't have the cold and rainy weather that firemen have to contend with in other parts of the country, there's no need for closed cabs on their trucks.