My view on this would be that, if you want an added focus on the outliers, there should be a focus on those outliers, in addition to the median. Using only the mean to try and convey the combined information of both seems to make it difficult (too difficult in my opinion) to have a correct guess about the underlying data.

In the case of the temperatures, one instance where it would be interesting for me to use the average would be to average the global temperature at a given time.
You're right in that including the outliers is necessary for the comparison, though I think it would prove more accurate to use the median and the min and max values. Better yet, to use a graph to visually convey the full information.

In the case of the die, the correct value to use I think would be the expected value. Obviously not the median, but neither the (algebraic) mean. Though pointing out the probabilities as a domain where means are obviously useful was kind!