This is by design, your door pad doesn't have a keyfob for your remote built in. If you want to use the keypad because you're running in to play ball and don't want your keys, just throw the keys in the console and lock the doors manually. No, its not armed but you get the convenience of using the keypad and not having to bring your keys with you.

Think of the keypad as an added feature. You're replacing your regular remotes with alarm remotes. Same thing here.

 

I had your issue #3 when I installed a viper alarm in my Titan. It could be related to all your problems too.

The fix was the alarm module has 1 output with a negative pulse when you arm/disarm. I forgot what it's called.

I wired it to the door pin switch so when you press the remote it turns triggers the pin thinking someone opened and closed the door.

If you have Amp Research automatic steps then you would need to run a diode to it because this pulse would also activate the step since that was wired to the door pins too.