Does the host nation flag fly at half-staff/mast with the American flag?
No. Not always.
For September 11, only America was attacked and only America flies it’s flag at half-staff. Other nations can authorize their flag at half-staff, it just depends.
See here: http://www.thedrillmaster.org/2012/07/16/the-american-flag-at-half-staff/.
Picture courtesy www.maggiesnotebook.com
Are state flags flown at half-staff on September 11th?
Yes, check your state’s guidance. Here is an example from www.state.md.us. State flags do not have to be flown at half staff when the American flag is flown at half staff. The American flag flown at half staff is a sign of mourning, during that time it does not matter that another flag is higher because of the purpose of the mourning position.
Half-staff is the term used on land. Half-Mast is the term used at sea or on land in the Navy/Marine Corps.