A/C goes on and off while driving, alarm siren no longer goes off when triggered, has ability to be shut off and keys removed while in Reverse?

I purchased this truck only 2 weeks ago, 2005 Nissan Frontier SE 4x4 automatic transmission.  First morning after purchase I noticed my key fob would not unlock the truck, so I used the key to manually unlock it -This set off the truck alarm. Horn honking, lights flashing, and the inability to get the truck to start. I soon realized the key fob had no battery in it ( I had been given 2 sets from the dealership and the other set was the one they had used on the test drive). After 20 minutes of tinkering with the ignition and auto door locks in the drivers side door I was able to get it to start with the alarm hazard lights still going - I drove into town like this and purchased a key fob battery and was able to get it to disarm (Husband had other key set). Since this occurrence however the truck no longer makes "alarm noises" when triggered, only has lights flash on and off. Also the  "beeping noise" to indicate I have locked or unlocked my truck via key fob no longer sounds. Whats up with this?

Second issue: My A/C will turn off and only blow cool air several times a trip. I have to shut off all heating/cooling system options and turn it back on to get the A/C on. Is it just overheating or is this an issue?

Lastly, I had come home one day and shut off, gotten out of the truck -keys in hand, and gotten inside before I saw my truck rolling backward in the driveway. I cam to find the truck was in "R" reverse in which I quickly put into "P". I have never had a vehicle have the ability to be shut off and have the keys removed from the ignition while the vehicle is in "D" or "R". Is this normal or yet another thing to have fixed?

Please Help. My dealership is not wanting to address this issues and I would like to take it elsewhere if these are legitimate malfunctions.

Thank you!