Is it possible to upgrade the iMac G3 (1998) to an iMac with Mac OS X Lion or Mountain Lion ?

A G3 iMac cannot be upgraded to run Lion or Mtn Lion; they are limited to OS 10.3.9 or OS10.4.11 depending on the hardware config. And Apple doesn't do trade-ins. Your best bet is to do a private sale of the G3 and apply what money you can get to a new Mac.

The 1998 iMac predates the introduction of Firewire. So yes, it is limited to 10.3.9. Sonnett did offer an upgrade to the iMac to allow it to have Firewire. I don't know if this enabled 10.4 or not. XPostfacto also offered unsupported 10.4 for older iMacs. But Spotlight and Dashboard are very slow if you can't upgrade to a minimum of 256 MB of RAM, and preferably 512MB of RAM. Look at http://www.everymac.com/ to find out your Max.

No, Snow Leopard - Mavericks requires Intel, G3 is PowerPC