Hopefully the moderator (Bob) will give you hardware assitance but I can tell you this:
1) Most notebook batteries need to be replaced after 2-3 years so if that is happening (low battery life) it is not necessarily due to the other problems it is due to the fact the battery is worn out.
2) No, you should not spend $500 to repair it. The Pentium III will not have similar battery life (and even built in wi-fi) like the new Pentium M Centrio systems have (and they weigh less). And you can get one for under $1200 and even closer to $1000 if you order with a student discount.
3) Dell has an outstanding reputation for DESKTOPS but as you are finding out their notebooks do not meet the same standard. Dell notebook keyboards are not responsive (especially compared to IBM notebooks) and their case construction also is bulkier and more brittle.
3) with that said, nearly all PC notebook manufacturers have notebooks made under contract overseas but there are better case construction choices than Dell. For example, the 8600 is very competitive for specs but it is heavier and does not have the keyboard fell or lower weight (and thinner) of the Compaq X1000/ZT3000 or even the Toshiba M35 series.
4) if you are having shutdown problems perhaps your hard drive connection has become loose or pulled out.
At any rate, if you are going to college now get a new notebook with built in wi-fi (with 54G) and better battery life with less weight.
Get an Intel Pentium M (Centrino) notebook or go Apple.
I have an Inspiron 8100 from a few years back. It has a Pentium III processor with 1.0Ghz. Also has 30 gigs and 512RAM. I have had many problems with this notebook. First, my battery is suppposed to last about 2.5 hours, but it only lasts 20 minutes. I called up Dell and they said that I need to replace my motherboard, which would cost about $500. I'd rather buy a new laptop than spend $500 trying to fix a crappy one. Second, all the space around my keyboard creaks. If I press it, it goes in and I feel that it is going to break. There are a few cracks not on the LCD display, but right next to it. The whole laptop is made of cheap material. And finally, my last problem, everytime I turn on my computer, a blue screen appears and it says "a problem has been detected and windows has been shut down to prevent damage to your computer" It also says UNMOUNTABLE_BOOT_VOLUME. I don't know what to do. I can't even use the computer anymore. I wanted to know if anyone knows anything about any of these problems and can help me. Has any of these things happened to anyone else before? Please help!