’16 Town & Country Alarm Triggered By Hatch

2016 Town & Country, power sliding doors/hatch, factory keyless entry but no factory remote start.
"Pursuit Pro 9233E" module added for remote start and alarm, not sure what else they used with it to do the install.

There are actually three problems I have with this setup:

1. If the alarm is armed, and we use the keyfob to open the hatch, the alarm is triggered. If the alarm is armed but we use the fob to open either of the sliding doors, the alarm is not triggered.
2. rapidly double-tapping the unlock button triggers the alarm - perhaps it's panic mode? (according to the 'manual' they gave me for the module, it should require unlock/lock/unlock sequence to do that). If I tap unlock once, then wait like a half second, and then tap it a second time to unlock all the doors, the alarm does not go off.
3. occasionally, when the alarm is going off, it does not disarm until you've tried disarming and re-arming a few times.

We've had the van for about a month and a half, and we let the dealer sell us on this add-on alarm. I was giving it some time to get used to its quirks, but also to determine what exactly wasn't working the way I expected, and what I could discern was causing the bad behavior.

I'm not an installer, honestly, but I'd like to know if the above things I've noted are typical for this module, or if this sounds like a simple fix that I can take back to the installer and have them resolve. Or maybe there's something wrong with the module altogether?

van has a 'fobik' key, is not push-to-start.

Regardless of whether it is a messed up installation or a hardware issue, none of what you describe is acceptable IMHO.
I would go back to the dealer and have him make it right.