If you are buying "fix a flat" look on the label for warnings of flammability. Some of the brands use a flammable propellant which means for the tire installer that the tire is now something that will go boom.

There is a product called slime http://www.slime.com/ They have a smart spare that has a can of goop and an 12 volt air compressor to go with it. I have used slime on a wide variety of off road tires and have had good results with it. It is a little more difficult to use than "fix a flat" because it requires unscrewing the valve core, squeezing the contents of a plastic bottle into into the tire, reinstalling the valve core, and re-inflating the tire.

Something else i have used with good results is the Safety Seal brand tire plugs http://www.safetyseal.com/store/autokits.htm
These require even more effort but it makes a very durable repair.

How does one go about telling if it is dead or not? I run all weathers in late spring and summer, and put winters on in late October, so the real spare gets swapped at the same time. I also don't put a lot of miles on my car, my commute is fairly short- should I be worrying about tire age as well as road miles?

I'll look to see if I have any pictures scanned of my Grandfather's "bush truck" when he worked in Liberia- he had a rack of four spares, with jerry cans of fuel and water under it on his pickup. Although those roads were only barely worthy of the name.

The local automobile club recommends changing the tires after five or six years. With a really short commute that may be an issue.
I know some people who use winter tires all the year because they still have to change when the time is up. Be aware that they require more room for stopping in summer (about 10%).