Welcome and glad to hear you were able to purchase the car.

Some of your issues that you listed could be a result of bad repairs. I would take the door panels apart and see if everything is plugged in and that the vapor barriers are secure.

As far as the alarm, it seems like the visual part is turned on but you do not have the acoustic part is not working due to the car not having the siren/sensors installed. And the rain sensor issue could be that the windshield is the cause since the cheap brand glass for some reason have issues with working ( or maybe its unplugged behind the mirror? )

Enjoy the ownership