Are you sure that the phone won't read the memory card? I'm not certain as it is my daughter who has one of these phones, but I believe that the phone won't function properly without an sd card inserted. If there is a problem with the memory card, have you tried putting it in a pc card reader to check if the card is okay, or a different card in your phone to see if that works?
I think that it is also necessary to install the drivers before you can use with a pc. When you connect the phone it should try to install drivers. If nothing is happening when you connect, have you tried a different cable as it could be that which is faulty?
Drivers can be found here: LG Product Support for GW620
If you do have a problem with the card, then it's unlikely you will get anywhere as usually connecting to a pc just mounts the memory card to the pc, rather than the phone memory.