1.Statistics deal with column data "distribution" in tables. Correct?
Yes
2. They are used (along with indexes) by the query optimizer to determine the best execution plan for a query. Correct?
yes
3. Statistics aren't updated automatically. How does one know when a statistic needs to be refreshed (e.g. using sp_updatestats)?
Depends on your database setting, if Auto Update stats option is On then sql automatically update stats.
http://www.sqlskills.com/blogs/erin/understanding-when-statistics-will-automatically-update/
https://www.simple-talk.com/sql/learn-sql-server/statistics-in-sql-server/
4. If a column has statistics defined on it, the table content is deleted and repopulated, will the statistics be automatically refreshed or will the old statistics remain intact?
Please read the above two links
5. I don't recall ever creating a statistic (unlike indexes) but the system seems to generate them automatically ("_WA_Sys...."). What triggers it to generate them?
not by trigger, sql creates them . Your database's auto create stats option is on, so sql create them automatically
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