Yes, CEA is paid only to a ceiling, or to 90% of fees whichever is lower.
State boarding is a very frequent choice for Forces families, who get some priority in admissions, which are based on suitability for boarding (the prospective pupil) and need for boarding (the family circumstances, where those in the Forces and those posted overseas by other government bodies, are top of the list)
CEA is not payable other than for boarding schools, and the pupil must board (and the rest of the family must accompany the serving person on each move). Occasionally, if serving person is posted to close to the school, the pupil can live at home for the duration of that posting, but they must have been a boarder before that). Occasionally it can be paid as a form of "board and lodgings" allowance if a DC stays put to continue at their school, living separately from their family but not in a boarding school - I don't know much about the ins and outs of the scheme in that form, but it's sometimes used when there's an unexpected move at Exactly The Wrong Time (is there any other?) icw GCSE or A level preparation.
Lots of people don't board their DC even if they would qualify for the allowance; there are a lot of personal factors that go in to the choice