Any fat can self-ignite at sufficiently high temperature, so your fear is in principle not unfounded – however, all common cooking fat/oil have a kindling temperature high enough that it should never be reached during normal shallow-frying. In particular, as long as you have any significant amount of moist food in the pan, evaporation will prevent the temperature from going much above 100°C. I.e., the more food you have in the pan the safer it gets, actually.
At lower temperatures, oil is hard to ignite, unless soaked into fibrous material. So especially refilling the oil in the middle of cooking is generally safe.
Stirring the food often required, and not unsafe per se. What can be unsafe is if fat splashes onto hotter surfaces or into a gas burner, but unless you're doing it in an extremely violent manner any splashes should be small enough to not cause disaster.
For e.g. steak, you may as you say need high heat for the right crust, and that can indeed lead you in risky territory: to get to that heat, you need to pre-heat the pan without anything in it, and then the temperature can easily reach autoignition range. Here, for safety:
- Don't use more oil than necessary. 2cm is way too much for frying steak. A layer that just covers the bottom of the pan will do.
- Know your stuff. An infrared thermomenter gives best control, but with some experience you'll already know how long the pan needs to get to a certain temperature at a given setting.
- Be prepared. If the oil does ignite in the pan, first of all don't panic. Next, shut down the stove entirely. If it's a ceran stovetop, carefully slide the pan to the coldest spot on it. Then cover the pan with a lid, with another pan of matching size, or with anything else solid that can't melt. That will cut off the oxygen supply and thus quickly squelch the flame without problems.
Never put water into burning fat, since it will violently expand and spray the flaming oil all around. That's the only thing that could really cause a quick disaster here!
