They are all state schools. Grammar schools are fully academically selective state schools. As opposed to state comprehensive schools, which admit either with no regard to aptitude or ability, or may admit a certain proportion of their intake through academic or other specific ability or aptitude, e.g.in sport or music.
For 2023 entry for Handsworth Girls, it says in the table on the page linked to by a pp,
Cat 5, 217, 7705m
(all PP offered)
This means that all (if there were any) on time Category 1 applicants (Looked After / Previously Looked After) were offered a place, all on time Category 2 (PP in catchment), all on time Category 3 (PP out of catchment), all Category 4 (in catchment, not LAC/PLAC or PP), plus some from Category 5 (out of catchment, not LAC/PLAC or PP), down to a score of 217, sub-ranking on distance 7705m (so not all out of catchment on 217 got in).
So for the current year, a non-PP score of 213 would not have secured a place, even if on time, but any PP applicant, applying from in or out of catchment, would have done.
The admissions policy for KEVI Handsworth Girls is here:
https://kingedwardvi.bham.sch.uk/wp-content/uploads/2023/03/Handsworth-Girls-2024-admissions-policy.pdf
Is your home address in one of the electoral wards listed? If so, this is your catchment grammar school and your DD will be Category 2, if you provided evidence of her PP status.
Every school has its own determined admissions policy - the Camp Hill one is like Handsworth's, but with different electoral wards in its catchment. Although it is also a KE Foundation member, Sutton Coldfield's is different (no catchment area, so no 'Category 2/3' or 'Catgory 4/5' differentiation). Queen Mary's has a slightly different policy again: no catchment area, but within the PP category, priority given to applicants attending a state primary school within Walsall.