from your symptoms - take the battery to an autoparts store, have them test it for dead cells... usually it is free... get a new battery, about $90 with 3 year warranty... this will eliminate some of the concerns, then you can move further in troubleshooting...

alarm is triggered when the battery is about to die, it also triggers engine kill switch.. when new battery installed, you cycle alarm with key, and then you can start the engine... last ditch effort to prevent theft...