IMO, I doubt it. The loose interior handle is just caused by a spring in the interior handle assembly, and shouldn't affect the door latch itself actually closing, which is where the problem would be. If you're not getting a 'door ajar' warning when you get in your car, then that's probably not the problem.
When you lock it, does the little red light next to your front door locks start flashing? If it does, then your alarm is being turned on, and as someone said before, maybe a previous owner just disabled the audio/visual part of the arming, so it's no big deal, really.
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