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If the vehicle will start, then there is not a theft system issue. The alarm going off while parked, if not an aftermarket added alarm, then its normally one of the door ajar switches or hid switch not closing fully and intermittently showing open. For the key stuck. the park switch is not releasing the ignition lock. Below is a link to the diagram.
The floor shifter park switch needs to use the BK black wire and ground the DG/OG dark green/orange wire when not in park, when in park the DG/OG wire is not grounded. Check this to see if the park switch or ignition lock is to blame.
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