When is the flag automatically flown at half-staff? Are there set days?

Yes, there are five days when the flag is flown at half-staff automatically.

1) May 15 — Peace Officers Memorial Day: from sunrise to sunset (section 136 of title 36, U.S.C.).

2) Memorial Day — According to the US Flag Code, flags are half-staffed until noon only, at which point they are raised to full staff.

3) September 11 — Patriot Day. Display your flag at half-staff from sunrise to sunset. On December 18, 2001, President Bush signed Public Law 107-89 designating September 11th as Patriot Day. The people of the United States are asked to observe Patriot Day with appropriate programs and activities to honor the individuals who lost their lives.

4) Fire Prevention Week — Sunrise to sunset. It is a tad complex. For Memorial Day and September 11, the legislation includes specific instructions on when to half-staff the flag. With Fire Prevention Week, it is different. The legislation simply says to designate that week. Then, the President issues a proclamation saying which date the flag gets half-staffed.

5) Pearl Harbor Remembrance Day — Sunrise to sunset, Public Law 103-308. It was signed into law in 1994 by President Bill Clinton.

An interesting observation on Pearl Harbor Day: many federal offices do not lower their flags on this day. This is not because they are wrong. It is because for Pearl Harbor Day, as with so many proclamations, the language says the President "invites" the people to participate. Federal agencies are ordered to participate. Private citizens are invited. The President has no authority to compel private citizens.