No. The GoLogic™ is a "dumb" USB device entirely controlled by your computer and cannot initiate I/O with any other device. The GoLogic returns data in response to USB commands from your computer. Only the computer physically connected to the GoLogic via USB can control it.
 

Does captured trace remain inside the GoLogic after power is disconnected?

No. The GoLogic™ uses high-speed volatile memory chips which lose all data when the 12V power supply is disconnected. Even the USB and logic analyzer programming are lost when power is disconnected. The GoLogic software loads these internal programming into the hardware at startup.

The GoLogic does not use USB power. Leaving the USB cable connected while the 12V power supply is disconnected will not retain the captured trace data or the logic analyzer's internal programming.