I just need to try out and register the motion sensors using the serial number / port number (1-5) to pair them with GC2. Am I correct?
Yes. You can find set up instruction for the TAKE on a GC2 here:
Instruction manual for the TAKE-345 can be found here
regarding sensors/camera, how do I reenroll them to the new panel? Do I need to remove the sensor from the sticker and press some button, similar to how I’d do that for Blink cameras?
Cameras do not typically pair to the GC2, what cameras are you referring to? Only Alarm.com compatible cameras can be used with service.
GC2 programming guides can be found here:
Assuming there is no way I can fix this without knowing the installer code, what are my options?
You need the installer code to access programming. The default code is 1561.
If I purchase a new compatible LTE module and install it, can I use Surety without knowing the installer code on the panel?
Yes, but you need the Master User code (default 1111) to run a cell test. A cell test is necessary to jumpstart communication between the panel and service.
To run a cell test, navigate to Security → Menu-> Toolbox → Cell Phone Test