Okay, you have an ATI Radeon HD 2400 Pro video card. I will admit right off the bat, that I have a major bias regarding this card because I used to have one in my PC. I had nothing but troubles with it as well. Similar crashing problems, that seemed to be pretty random.
The problem with this card is that it is not really a "gaming" card. It is more of a mainstream card. It does have cool features, like displaying HD content on your PC to your TV. It does have 256MB of video RAM, but because the memory interface is only 64bit, you can't really take advantage of that. It also has a pretty low memory and core clock speed.
You can definitely update your video drivers. ATI has a driver dated 10/15/08. Your current driver is 3/29/08, so that mighthelp.
Next issue - is system RAM. You have 1GB, which is generally okay for Civ, but how old is your PC? Your BIOS is dated 2004, so is your PC 4 years old? If so, you may have so many processes and programs running in the background that even if Civ 4 is the "only" program running, your system may be simply running out of RAM after you have been playing for a while.
You could try killing some of the extraneous programs and processes to see if that helps. Additionally, you might try enabling the line in the config file that says:
; Allows some memory savings *** ALT-TAB WILL NO LONGER FUNCTION ***
MemSaver = 0
(Change the "0" to a "1")
You should also try messing with some specific in-game settings besides "low" graphics settings like single unit graphics. You could also try playing the game in a lower resolution (like 800x600). This will put less pressure on your graphics card. Finally, you can try playing mostly smaller map sizes. The smaller the map size the less graphics and memory intense the game will be.
Good luck! and post back with updates/further questions.