Sorry for the long post, the bullshit asymmetry principle applies here. Possibly someone else has explained this better in fewer words in which case also sorry for x-posting.
But these exemptions apply to all the protected characteristics in the act. So if it could be shown to be a proportionate means of meetings a legitimate aim it would be legal to discriminate against lesbians and gays - and some religious employers do, often quite dubiously under the genuine occupational requirement exemption. Is this a sexuality based right? Is it a 'straight' right?
No this is not how the EA works.
It's unlawful to directly discriminate against someone because of a protected characteristic unless you are lawfully using one of the specific exceptions in the EA, including those written into section 13, each of which deal with specific protected characteristics and specific circumstances.
www.legislation.gov.uk/ukpga/2010/15/section/13
Indirect discrimination (same rule for everybody but people with a certain protected characteristic are disproportionately adversely affected by the rule) is only lawful where it is a proportionate means of achieving a legitimate aim. Indirect discrimination does not apply to the PC of pregnancy and maternity.
www.legislation.gov.uk/ukpga/2010/15/section/19
There are separate sections that deal with pregnancy and maternity discrimination in work and non-work cases, discrimination arising from a disability and gender reassignment discrimination regarding absence from work.
There are separate sections that deal with the duty to provide reasonable adjustments for disabled people. No other protected characteristic is entitled to reasonable adjustments.
Harassment is prohibited conduct but this section does not apply to the PCs of marriage and civil partnership or pregnancy and maternity. Victimisation is also prohibited conduct and applies to all 9 of the PCs. For both harassment and victimisation you do not have to have the PC yourself.
www.legislation.gov.uk/ukpga/2010/15/section/26
www.legislation.gov.uk/ukpga/2010/15/section/27
The rules are again different for different protected characteristics depending on whether the situation is to do with services and public functions, premises, work, education or associations and the particular circumstances that might arise in any of these areas.
The exceptions that permit direct discrimination are each specific to certain protected characteristics and not others, each exception relates to specific circumstances and each has its own legal tests.
So for example there are exceptions that say it is lawful for religious leaders to refuse to solemnise a marriage of a same sex couple or if they believe one of the couple has changed their legal sex. They don't have to show that this is a proportionate means of achieving a legitimate aim. They can just say no.
www.legislation.gov.uk/ukpga/2010/15/schedule/3/paragraph/24
www.legislation.gov.uk/ukpga/2010/15/schedule/3/paragraph/25A
There are no similar exceptions for registrars conducting civil marriages and no exceptions that make it lawful for religious leaders to refuse to marry eg. mixed race couples or people with disabilities.
For example there's an exception that permits an employer to only hire someone for a particular role if they have a certain protected characteristic. The rules here are quite convoluted.
Outside of religion, the employer needs to show that not only is this a proportionate means of achieving a legitimate aim but also that it's a genuine occupational requirement (e.g you can't only hire people of a particular race because they are underrepresented, you also need to show it's necessary for that particular role).
For 7 of the 9 PCs the requirement is that the person has the PC (e.g. it's lawful to require an employee to have a specific disability but not to require them not to have a disability) but for gender reassignment and marriage and civil partnership the opposite is the case. You can require employees to be single or not trans but you can't require employees to be married/civil partnered or to be trans. Frankly I don't know why TRAs are not focusing on this part of the EA which makes it unlawful to specifically hire trans people e.g. for specialist counselling roles. It's almost as if this isn't about the rights of trans people at all.
www.legislation.gov.uk/ukpga/2010/15/schedule/9/part/1/crossheading/general
When it comes to organised religion there are a whole other set of rules. It is lawful for organised religions to only hire people of a particular sex, only hire people who are not trans, only people who are not married/civil partnered, only people who are not in a same sex marriage/civil partnership, only people who are not divorced/who have dissolved a civil partnership or only people with a specific sexual orientation. They are not required to show a proportionate means of achieving a legitimate aim. Instead, the test is whether it would conflict with the strongly held religious beliefs of a significant number of their congregation. Religious organisations still cannot discriminate on the grounds of age, disability or race.
www.legislation.gov.uk/ukpga/2010/15/schedule/9/part/1/crossheading/religious-requirements-relating-to-sex-marriage-etc-sexual-orientation
For example there are exceptions that make it lawful to provide separate or different services for males and females, or to only provide services for one sex. All of these require the service provider to be able to show that their use of these exceptions is a proportionate means of achieving a legitimate aim.
These exceptions are specific to sex and no service provider will get anywhere if they try to extrapolate these exceptions to other protected characteristics. Try segregating on the grounds of race and see how quickly you end up in court.
There is a related exception that makes it also lawful to exclude people with the PC of gender reassignment when using one of these sex based exceptions.
Again, religious organisations have different rules and do not need to meet the test of showing a proportionate means of achieving a legitimate aim. They just need to show it's against their religious doctrines or would piss off a significant number of the congregation.
www.legislation.gov.uk/ukpga/2010/15/schedule/3/part/7
TL;DR: Exceptions in the EA are exceptional and each exception has specific rules around which PCs it applies to and in which circumstances. It's not a fucking pick-and-mix.