Seek your help as new problem has emerged in my computer.

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My computer is making troubles again for me. Previously it was running slow but with change of AntiVirus device it recovered speed. But now it has started creating new problems for me. It gets hot within a few minutes of starting. At first I thought the fan within was not functioning/working but soon I discovered that when the Internet is ‘on’, it gets heated but when the Internet is ‘off’, it cools down gradually and if I do not open the Internet, it never gets heated. Can anyone explain the cause behind and suggest any remedy for it? Another new problem has emerged recently for which I also seek your kind help and advice. While working online a message is often displayed which I am quoting hereby- ‘Generic Host Process for Win32 Services has encountered a problem and needs to close. We are sorry for the inconvenience’. And all the online connection goes off. The icon of online connection remains as it is but no site is available. If I click on any site, a message is displayed stating that the site is unavailable. As I use mozilla firefox, I am quoting hereby the message displayed by it- Firefox can't find the server at www. *******.com. Check the address for typing errors such as ww.example.com instead of www.example.com If you are unable to load any pages, check your computer's network connection. If your computer or network is protected by a firewall or proxy, make sure that Firefox is permitted to access the Web. In this context, I like to mention that my computer is old enough as it was bought in the year of 2003. So should I abandon it as a condemned one and purchase a new one to get rid of these occasional harrasments or try to continue with it as long as it is possible?