people can't change their legal sex until 18
Yes
thus sex as a protected characteristic prevails in cases where there is a need for single sex provision.
I wouldn't phrase it like this, but sort of. It's not that the PC of sex prevails, it's that the notional transgirl's sex is male as they don't have a GRC. So when you are having things like segregated changing rooms, you exclude the transgirl along with all the other males from the girls facilities.
The law allows for 2 genders thus the idea of a male/female GI exists in the legal system but goes no further (i.e. 99 genders).
Sort of. There are 2 legal sexes/genders, but sex and gender almost always mean the same thing in law (indeed, they meant the same thing everywhere until the notion of gender being the social aspects of sex was developed around the 1950s/60s. Gender had been used for hundreds of years up to that point as a synonym for sex).
However, children who plan to transition will engender a protection under the EqA.
Yes.
What would that look like in a school? Misgendering is classed as harassment? Deadnaming, is this allowed?
That's hard to say as there have been no cases on this. This is what the new draft guidance for schools aims to cover.
If the child needs to leave school to go to the endocrinologist, should this absence be marked as an acceptable medical reason?
Yes
Can a girl object to a boy participating in a single sex PE class e.g. badmington (i.e. non-contact).
Again, it's hard to say. They definitely need to change separately. We did mixed sex badminton at my school (bog standard comp) when I was 15 or 16 anyway, I don't know what other schools do. For contact sports there are obvious safety concerns. I personally don't feel girls should be disadvantaged competitively in non-contact sports but the legal position is not straightforward.
Naively this sounds to me like gender and gender identity beliefs are being validated by our legal system.
I don't think GI beliefs really were a common thing when the legislation was drafted, and as above gender just means sex in law in almost every single context, so I wouldn't read that into it myself.