The way I read it: The period from August 2005 through July 2015 is removed, ASSUMING that he was in the military for all 10 years, so he had 36 months from July 2015 to have a 60 month look back that included 24 months of residence. If he sold it before August 2018 (we don't know the actual day, and it doesn't matter).

He sold it September 2018. Too late to claim anything. In the 60 month test period he lived in it 22 or 23 months. No exclusion.

The military suspension is for the 60 month test period. Since he lived in it for many months through July 2005 (more than 24), 24 of those count in the 60 months. Then he left, but the test period is suspended while he is on qualified official extended duty. IF that lasted until July 2015, then the test counting begins again. 60 - 24 = 36.