Alarm activated when it wasn't engaged. Now car won't start runn

The Lock's lock from the door switches ? If they do it's not a fuse problem . Which of these does your vehicle have ?
Keyless Entry System Description and Operation (with Accessory 2 Way Remote)
Keyless Entry System Description and Operation (without Accessory 2 Way remote)

The keyless entry system is a vehicle entry device. The keyless entry system is used in conjunction with the body control module (BCM) to remotely activate certain vehicle features. Keyless entry will lock/unlock the doors/liftgate when a corresponding button on the keyless entry transmitter is pressed. This is accomplished by the transmitter sending a radio frequency to the remote control door lock receiver (RCDLR). The RCDLR interprets the signal and activates the requested function via a serial data message to the BCM. A low transmitter or vehicle battery or radio frequency (RF) interference from aftermarket devices, such as 2-way radios, power inverters, computers, etc., may cause a system malfunction. High RF traffic areas may also cause interference that could lead to a malfunction. Keyless entry allows you to operate the following components:
• Door locks
• Liftgate lock
• Vehicle locator/Panic alarm
• Illuminated entry lamps
• Remote vehicle start
The keyless entry system has the following components:
• Keyless entry transmitters
• Keyless entry antenna
• 2 Way Advanced Remote Start RCDLR
• BCM

Your best bet , take it to a GM dealer .

Based on all the info you have provided I think you need a professional to get involved here, I am not willing to "GUESS" what might be the issue or issues (more likely more than one).
Good luck

Go on youtube and put in how to reset your car key after renewing key fob battery good luck colin

Testing would need to be done , scan tool hooked up checking DTC'S - diagnostic trouble codes . View data etc...
Keyless Entry System Description and Operation - Active
The keyless entry system is a vehicle entry device. The keyless entry system is used in conjunction with the door locks to unlock the vehicle. Keyless entry will lock/unlock a door or open the rear compartment lid when a corresponding button on the keyless entry transmitter is pressed. This is accomplished by the transmitter sending a radio frequency to the remote control door lock receiver (RCDLR). The RCDLR interprets the signal and activates the requested function via a serial data message. A low transmitter battery or radio frequency (RF) interference from aftermarket devices, such as 2-way radios, power inverters, computers, etc., may cause a system malfunction. High RF traffic areas, such as gas stations that use pay-at-the-pump RF transponders, may also cause interference that could lead to a malfunction. Keyless entry allows you to operate the following features:
• Door lock/unlock
• Rear compartment lid release
• Illuminated entry lamps
• Panic alarm/Vehicle locator
• Remote vehicle start

Your best bet , take it to a qualified repair shop . Dealer

Many times the dealer gives you a valet key made to give to the vet to park your car, these keys work even though the theft deterant system is active, but they will only work for short periods untill the theft system activates and imobolizes the vehicle. The system is reset when the valet brings your vehicle and you insert your normal (chiped) key.

Is this an issue with all 2008 HHR remotes??? Highly doubtful , probably the vehicles keyless entry system . Taking the vehicle to a qualified repair shop an having the keyless entry system checked for DTC'S - diagnostic trouble code's . Do you know what a DTC is ?
The keyless entry system is a vehicle entry device. The keyless entry system is used in conjunction with the body control module (BCM) to remotely activate certain vehicle features. Keyless entry will lock/unlock the doors, or release the rear compartment lid when a corresponding button on the keyless entry transmitter is pressed. This is accomplished by the transmitter sending a radio frequency to the remote control door lock receiver (RCDLR). The RCDLR interprets the signal and activates the requested function via a serial data message to the BCM. A low transmitter or vehicle battery or radio frequency (RF) interference from aftermarket devices, such as 2-way radios, power inverters, computers, etc., may cause a system malfunction. High RF traffic areas may also cause interference that could lead to a malfunction. Keyless entry allows you to operate the following components:
• Door locks
• Liftgate lock
• Remote rear doors (Panel only)
• Vehicle locator/panic alarm
• Illuminated entry lamps
• Remote vehicle start
The keyless entry system has the following components:
• Keyless entry transmitters
• Keyless entry antenna
• 2 Way Advanced Remote Start RCDLR
• BCM

Guessing will not fix this .

The 'Keyless Entry & Burglar Alarm System" is activated. You must 'Disarm' the system using the Keyless Entry Remove (Key Fob) to "Reset" the system. The reason the engine won't start is due to the burglar alarm - it has a "Starter Kill" circuit which is activated when the alarm is activated. The alarm must be DEactivated to DEactivate the 'Starter Kill' circuit.
Another way to start the engine is to get in the car, put the key in the ignition and turn it to 'ON" and wait 3 to 5 minutes until the emergency flashers stop/turn off. Afterward, you should be able to start the engine. NOTE: This does NOT reset the burglar alarm. It will go off again when you enter the vehicle without using the remote/key fob.

what do you mean keyless entry? don't you have a remote unlocker?

check the releys under the hood play musical releys and find the problem child
munber all the releys 1 - 10 then start playing musical releys

WITH KEY IN DRIVER'S DOOR LOCK/UNLOCK DOOR TWICE. THIS SHOULD OVERRIDE ALARM SYSTEM. STRONGLY SUGGEST BUYING KEYLESS ENTRY FOB.