1. Statistics are a very small data set representing the selectivity/density and the histogram, or distribution, of the first column of the data defined by the statistic. This is true if it's on an index or just a set of statistics created automatically on a column referenced in a filtering clause in T-SQL code. It doesn't rearrange data, so no sorting is done. It does read the data, either in a sampled fashion, or by scanning the entire table for the column(s) defined by the statistic. Statistics are part of indexes, but they are separate from them.

2. Just about every process hits tempdb to one degree or another. I wouldn't be surprised if this one does too. But, again, the statistics are a very small data set, 200 rows, so updating them usually is not a major process unless we're talking hundreds of millions of rows of data.