In short, for the question of preventing injury, it's entirely relative to one's current fitness, nutrition, age, and current physical status. There is never a guarantee that stretching and warming up will prevent injury. The point of warming up is to limit the likelihood.
For example: if one in relatively good shape and is relatively young and have no physical limitation such as a leg cramp, then in the cases you mentioned the risk of injury, in exercising with out warm-up, is relatively low.
Conversely, in the other extreme, if one is relatively old, over weight, never works out, eats unhealthy, and currently has a leg cramp, then injury would be highly likely to occur.
In both cases, however, injury is neither guaranteed to occur or not occur. Sorry the answer is not definite but in this case one cannot speak in absolutes.
That said, Yes, it would be helpful in preventing injury to increase your fitness in a given activity in spontaneous things like running short distances before engaging in them to better condition your body. Spontaneous efforts will, at first, engage the fast twitch muscles so you'll also want to add a strength training regiment. Strength training will benefit you most for explosive efforts.
If your ultimate goal is simply to do it with less hard breathing and sweating then cardio/pulmonary workouts [i.e. exercises that strengthen the lungs and heart] can increase lung and heart efficiency. Strength training is great for your heart but does little for lung endurance. This is where HIIT may become useful.
As for sweating though, arguably, if you have to exert less efforts to receive same benefit then your body will need to sweat less but, at the same time, the more fit you are, the more easily you will sweat. See this article.. Other factors come into play as well such as ambient temperature, hydration, etc. For the issue of sweating, you can wear clothes that well breathe well and dissipate the sweat.
HIIT is great for the lungs and heart but it is not a replacement for strength training and will not produce the same results in the short term or even long term. HIIT is the perfect method for losing fat while retaining muscle mass. It is also a good way to train for spontaneous activities especially if you also desire to burn calories. Some people define HIIT differently, so I'll add this qualifier Weight training being -i.e. weighted squats, leg press, calf extensions, lunges, etc. and HIT being non-weighted High Intensity Workouts. Some HIIT techniques involve weighted exercise but they're applied with a different methodology. The focus with HIIT is to burn calories not to build muscle strength. One of the drawbacks of HIIT is that many of it's activities may put you at risk of injury. If you're concerned about injury or have had a recent injury, I wouldn't recommend HIIT.
More on why HIIT is not a replacement for strength training:
“EMG techniques make it possible to study recruitment order, the relationship between stimulation and the amount of force developed, the type of muscle contraction (concentric vs. eccentric) and the effects of fatigue. EMG analysis in my study showed the approximate percentage of the recruitment of muscle fiber types in the quadriceps of a trained athlete during execution of a one repetition squat with progressively increasing loads.
Starting with 60% of one-repetition maximum, the slow-twitch fibers contribute 60 percent to the effort; fast-twitch fatigue resistant fibers, 30 percent; and fast-twitch fatigable 10 percent. At 100 percent maximum effort, however, the percentage of slow-twitch fibers involved is only 5%, while fast-twitch fatigue resistant is 15 percent, and fast-twitch fatigable is 80 percent.
The implications for athletic-type strength training are clear. To develop strength in the fast-twitch fibers you have to train with heavy weights. Light weights contribute little to optimizing strength and power performance. “
For complete article see: http://www.drdarden.com/readTopic.do?id=412352