Don't worry about being a pest, these are important topics.
Non-stick cookware can be very dangerous, but at the same time some of the warnings are a bit overstated. For the quick answer food cooked in non-stick cookware is perfectly safe for birds and humans.
The danger in non-stick cookware, specifically PTFE (ie Teflon) coated pans is that at a high enough temperature the coating will break down giving off toxic fumes. These fumes are very toxic and can quickly kill birds that are exposed to the fumes.
What is often overstated in the case against Teflon however is how often such fumes are released. Under normal cooking conditions the cookware would never reach temperatures sufficient to break down the PTFE coating. I don't remember the exact number, but I think it was in the range of 600 degrees F before any fumes are given off. Most cooking oil would ignite well below that temperature, which means if you have a teflon pan with cooking oil in it and you still have eyebrows then there are no PTFE fumes.
Is it still a good idea to change out non-stick cookware? Yes, for the most part it is but normal responsible cooking with teflon pans would not be a concern. All the horror stories of birds dying from these fumes involve someone putting a dry pan on a burner then falling asleep, going out for coffee, etc. And while accidents do happen if you ever leave open flames unattended in your kitchen then teflon fumes may not be the biggest concern.
Self cleaning ovens are coated with a PTFE coating or other similar material. These ovens DO reach temperatures well in excess of 600 degrees during the cleaning cycle - hence these ARE quite dangerous as they will give off fumes in their normal use.
I still have a couple non-stick pans that I use regularly. I never 'preheat' a dry pan, I don't use them in the oven, and I never leave them unattended (while cooking). I did get rid of my George Forman grill as these are preheated dry for normal use. George Forman style grills, pancake/waffle irons, and other such appliances are a concern while other teflon cookware is safe as long as its used responsibly - but of course getting rid of them is the only sure way to prevent any accidents (forgetting about a pan on the stove, etc).