There would be a difference to the median. The old number wouldn't be the median but the mode wouldn't change.Mean? If the outlier is a high value, it will cause the mean value to shift to the higher...
I'm having trouble seeing the dataset clearly.
Yes. For instance, the dataset {1, 2, 2, 3, 4, 4} has modes 2 and 4.
It has no mode.
A set of data will not always have a mode because some data sets will not have a number that occurs more than once.