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