Torn tyres should be replaced ASAP. A boot will get you home, but a damaged sidewall is a blowout waiting to happen. An old worn spare will see you through till you can get a replacement, hence the usefulness of saving them.

Is there a difference between racing tyres and training tyres? I would imagine that weight is less of an issue for training, but toughness and longevity would be more important.

Winter tyres should be a little wider than summer ones. In the rain and dark, you need to absorb a little more abuse from the road.

I have found that tyres of the same nominal size vary considerably. My new 32mm Panaracers are thinner than the old Michelin Tracers they replaced (now sadley unavailable). Some tyre/rim combination can be a bit tight, and can lead to sidewall damage every time you remove them.