Adoption was only formalised after the Adoption of Children Act 1926 so is irrelevant in this case
Not necessarily - many adoptions that had happened years before the 1926 Act were taken through the courts afterwards, to make sure they were properly recorded (especially for inheritance rights for example) and that could be done up until the "child" was 21, so it isn't uncommon to find pre 1926 birth entries marked as "adopted"
Index of Adopted Children (a distinct and separate index to the normal birth index), is not open the the public.
The index for the Adopted Children Register (ACR) is available for anyone to search, and certificates can be ordered using it, but it isn't on-line. It can be viewed on microfiche at the designated libraries who hold copies of the GRO indexes ( along with things like the most recent BMD entries, indexes of civil partnerships etc.).
However the ACR doesn't give any link back to the original birth name/entry - that information is restricted.