Hi Julia,
1) Yes. If you are using en eprime script to handle the serial messages, both computers should be up and running. On the receiving side you probably need to put the ReadString function in a loop (often refered as polling). But see below for an alternative.
2) You are trying to send basic commands over the serial connection. In theory this would be possible if the receiving side could translate these strings into executable code (called evaluation). I'm not sure, but if I remember correctly, E-Basic does not have an 'Eval' function to do this. Also, even if the eval function would be available, I would strongly advice against using it this way. Since you only require a simple trigger signal, it is sufficient to send a single byte. Sending one byte instead of a complete string simplifies the receiving script significantly. You could use the ReadString method as described at 1). However, E-Prime also allows a serial port to be used as input device. The Serial input device will handle bit-changes as responses or events, just like the keyboard device. Because it's implemented as input device, there is no need to create a complex script that interprets the received byte stream. So, the receiving script should only include an object that waits for any event on the serial input device, and then continues with the next object to play the sound. The sending side could be really simple, with only two or three lines of inline script. However, you might have to use WriteBytes to send a single byte, instead of WriteString.
3) USB-RS232 conversion should not be an issue. (Although timing accuracy might be worse.) However, make sure to use a null-modem cable to connect two PC's.
http://en.wikipedia.org/wiki/Null_modem
Note: if both PCs have a parallel printer port, it might be easier to use those instead of serial/usb ports. Timing would also be more accurate in that case.
Best,
Paul