I added a carbon monoxide detector specifically to my bird room. Here's a video about that and the dangers of CO. $50 to let the plastic be the canary in the coal mine for the real birds is a done deal!
This is the one that I bought. It just plugs into the wall, has a 10 year battery, and has an LCD read out to let you know about trace levels that might not be sufficient to trigger an alarm.
Yes, very important to have them! I have the kind that are a combination of both smoke and carbon monoxide in every room - they are also the kind that are hard-wired to electricity but have a back-up battery. It pays to be extra careful!
Yep. You need one and also make sure you clean up any ducts you might have as there are services who eliminate carbon monoxide.I have tried that myself with vent holes and air condition in the house in order to keep a moist environment for them and have them growing faster. It works quite well I have to admit, but you need to be very careful with the carbon monoxide emissions if you don't clean your system regularly, or you don't even know how to do it. In Australia, we are like you since we have some really cheap and convenient services to call like https://supremeductcleaning.com.au/ who do a great duct cleaning in just an hour.