Can I somehow play the network version? I read from the manual, that IPX is needed.

Ignore the manual, since it is quite old. At the start menu you should have some shortcuts under "GOG.com\Carmageddon\Carmageddon\Network". One should select Host, and write down the IP address, while the rest should select Client and give the IP address of the host. If you play on a lan, the ip address will probably be something like 192.168.x.x or 10.x.x.x while if you want to play through the internet, you'll need to find the internet IP you are using and also forward the needed ports.
Unfortunately, I haven't played it multiplayer yet, thus I can't give any more help.