There is a widely held belief among brewers that yeast should be propagated at, or very close to, the same temperature as the subsequent fermentation. I have always thought that this general rule comes partially from empirical observations and partially from what seems to make sense. If you are getting ready to ride a bicycle race in Colorado, it makes sense to train in the mountains and not at sea level. So if you are preparing lager yeast to ferment at 58 °F (14°C), then it seems logical to propagate the yeast at 58 °F (14 °C) instead of 80 °F (27 °C). The problem I have with this logic is that comparing how an animal responds to environmental conditions is far more complex to how a single-celled organism like yeast responds to environmental conditions. We humans love anthropomorphisms, yet the connections are usually not so obvious.
One of the common misconceptions about lager yeast strains is that these yeasts like the cold and are not as happy when grown at warmer temperatures where ale strains seem so content. That’s three anthropomorphisms in one sentence and the temptation to think of yeast cells as little people is not particularly useful to brewers. The fact is that lager strains grow really well at warmer temperatures and it is also true that the resulting beer is fruitier and ale-like than when the same strain is fermented at cooler temperatures. Anchor Steam is an example of a lager beer fermented at warmer temperatures. The same is true for ale strains; reduce the fermentation temperature and fruitiness usually follows. The main difference is that lager yeasts can ferment at much cooler temperatures than ale strains, and when fermented at the lower end of their functional range, lager strains result in very clean beer.
In my experience, lager yeast can be propagated at warmer temperatures (68-77 °F/20-25°C) and then pitched into cooler wort and be successfully used for lager fermentations in the 50-55 °F (10-13 °C) range. During yeast propagation, brewers are more concerned about growing healthy yeast cells than controlling the environment in such a way to optimize beer flavor. In order to optimize growth the propagating culture is aerated using a variety of methods.
On a small scale, flasks stoppered with cotton plugs and mixed using stir plates or shaker tables have enough gas transfer at the gas-liquid interface to provide oxygen to the growing culture. In larger propagators, air or oxygen is bubbled into the culture and the transfer of oxygen into the liquid is often aided by the use of mixers. In all cases, oxygen is added during fermentation to benefit the growth of yeast. This practice is far different from normal beer fermentations where oxygen is usually detrimental to beer flavor (there are notable exceptions to this rule, such as York-shire Square fermenters). My point is that yeast propagation and fermentation methods have some key differences.
You ask about the potential for mutations if lager yeast is grown in a warmer environment. Again, human perception comes into play. Many of us think of a three-eyed fish as a mutation, but mutations are rarely so obvious with yeast cultures. Subtle changes are more common with yeast strains. Cultures may slowly become less flocculent over time, or diacetyl reduction may become problematic, or attenuation may begin to suffer as a culture is used over and over.
Brewers know that changing the environment, for example the fermentation temperature or wort original gravity, is one way to change how a yeast strain behaves. As brewers conduct a fermentation, harvest the yeast from the top or bottom of the fermenter (depending on yeast type and harvest method) after fermentation, re-use the yeast crop and repeat the process, we are selecting cells from the total yeast population for re-use based primarily on flocculation properties. Over time the culture may lose the properties that are desirable to brewers and when this occurs, the yeast is re-propagated from laboratory cultures. This is perhaps the primary reason commercial brewers want to grow the yeast in a similar environment that fermentation takes place. In my view of this subject, I do not believe the nuances of propagation temperature, within the norm of what most of us consider “room temperature,” are likely to have much effect in the typical homebrewery.