The factory immobiliser is a function of the ECU and BCM (inside, under the dash). Keyfobs have to be registered to the vehicle (there's a function to do this that can be done with workshop computer devices can do).
What are some of the issues? Usually - and I'm talking "commonly but not always" - multiple (and seemingly unrelated) issues are caused by electrical faults, most often just dirty battery terminals but sometimes also the ground cable connecting to the battery negative to the engine block (and rarely, the main fusible link which can crack and start causing weird problems). Other possibilities are the earthing of the stereo which if not "good", will cause the instrument cluster to do random things as the two share an earth.