I can never tell whether threads of this kind are merely seeking affirmation. With those predicted grades, and super-curricular achievements as described, you would be a competitive candidate. Whether you succeed in an application to Oxford would depend mainly on how you do in the aptitude test and at interview. Achieved and predicted grades, reference, and personal statement would also be assessed. The selection process is not a mechanical one of counting up how many A* grades a candidate has or is predicted to obtain.
Why be deterred by something being competitive? Not entering a competition is the surest way not to win that competition. You entered the Maths Challenges. Those were competitive events too.
Being middle class and attending a grammar school won't impede you. Most students who get into Oxford are middle class, and those from state schools have often attended high performing schools.
If you would like to study at Oxford, apply, choosing four other universities that appeal to you, including one safety option in case you blow up in your A Level exams.