|On some laptops, the fan does not come on while booting and only comes on after that when the cpu temp (or the temp inside the laptop) goes above some threshold.|
On other laptops, it comes on while booting then shuts off within a short time and doesn't come on again unless the cpu temp goes above some threshold.
It's never running all the time, unless where you're using it is really hot, or the air passages inside the base are clogged, or the openings that let air in and out of the base are obstructed (blocked) externally.
You can check the current cpu temp (or the temp inside the laptop) in the bios Setup.
If what you see there indicates the cpu is actually getting hotter than it should, and/or if you're sure the fan never comes on, then you need to replace the fan.
Search on the web for : Latitude D800 Service manual
and one of the "hits" will be that on the Dell web site.
The fan may have the part number you need to search the web with on it, or the part number may be in the manual.
How old is your main battery?
Laptop main batteries usually work as they should for a year or so, then they begin to rapidly deteriorate. From what I've seen, by two years they cannot be charged to anywhere near their full capacity, and it gets worse after that.
They often develop internal shorts when they are older. If the battery gets HOT rather than just warm after you have attempted to charge it for at least a half hour, or after you have been using the laptop for a while with the AC adapter plugged in, it's definitely internally shorted and you must replace it. Internally shorted batteries getting too hot have been known to cause the laptop to catch on fire.