Sex is the protected characteristic under the Equality Act 2010 and the only two possible options for sex are 'Female' and 'Male' as defined in the Act and consistent with biology. 'Other' is not a valid option.
You then ask, "Is your sex the same as the sexyou [sic] were assigned at birth?"
Sex is not 'assigned' at birth: sex is observed and recorded and is immutable. This question, therefore, makes no sense and it is not clear what information you are seeking to gather here.
3/5
Your job application mentions your statutory obligations under the Equality Act and correctly has sex in a list of the protected characteristics under the Act.
However...
1/10
However, you then ask "Please choose the option which best describes your gender identity" with options:
Male
Female
Transgender.
2/10
'Gender identity' is not a protected characteristic under the Equality Act 2010 and is not defined in the Act.
The Equal opportunities section of the job application mentions the Equality Act 2010 four times and correctly lists sex as a protected characteristic.
However, you then ask, "Please indicate your gender" with options:
Male
Female.
2/11
'Gender' is not a protected characteristic under the Equality Act 2010 and is not defined in the Act.
Your Recruitment Diversity Information questionnaire has 'gender' in what appears to be a list of the protected characteristics under the Equality Act 2010.
1/11
'Gender' is not a protected characteristic under the Equality Act 2010 and is not defined in the Act.
The Equal Opportunities Form in your job application has 'gender' in a list of what appears to be the protected characteristics under the Equality Act 2010.
Sex is the protected characteristic and the only two possible options for sex are 'Female' and 'Male' as defined in the Act and consistent with biology, but you don't ask for that.