Alarm system going off cannot stop it?

First make sure it is a factory alarm system. After-market alarms are a different animal and the best solution to them is to remove them and put them in the trash.
Now if it's a factory system, look in the left front of the engine compartment, there should be a hood switch that tells the alarm if the hood is being opened, unplug it and see if the alarm still triggers. If it stops the switch may be bad or more common is that it gets bent down, so it doesn't operate. Now if that doesn't work you really need to have a scan tool, with it you can look at the 4 door switches and verify that those are working properly, then you can also look at what the ECU thinks is causing the alarm to go off.
Do you have a known good key that is programmed in? Have you tried a second known good key?
As for resetting the alarm. Use the door lock system to unlock and relock the doors, then unlock them and then use a key outside the door to lock and unlock the driver's door. The factory system arms once the doors are locked and closed and should only go off if one of the doors is opened or the hood is opened. If you lock, then unlock and lock and unlock the door with the key, the system should be disarmed. If it doesn't then either the hood or a door switch isn't letting it disarm, or the ECU has an issue.