ping is to a certain server.
1. pinging the speedtest.net server(s) may well have very different values to another server you ping -- it depends where the servers are, their current load, the connection line quality, etc.
minor:
1. are they both to the same exact server? if so, it may well be you're on a different exchange or something, depending on how your connection is setup with your ISP (I have no idea how it works in your country though).
2. if it's very very old then yes. but yours isn't that old, so no, your computer's specifications will not be a factor of your ping times.
3. the viruses could be slowing your computer's connection in general, thus including speedtest.net.
AVG free is good:
http://free.grisoft.com
also get their spyware protection, or adaware. and make sure you're running a firewall like zonealarm free.