Can an employer offer a promotion without explaining what happens if you decline it?
Yes. In the absence of any other information, the assumption would be that she will remain in her current role on her current terms if she rejects the promotion.
Is a promotion optional, or can they pressure you into accepting it?
A promotion is a change of her employment contract which therefore requires her consent. They can make life difficult for her if she refuses, but they cannot force her to accept the promotion.
Is it normal for a promotion to come with no pay rise unless you increase hours?
No, but that isn't to say it never happens. Some promotions actually come with a pay reduction, for example when the employee is moving from a low level job with variable hours which can include lots of overtime to a management role where paid overtime is rare.
Is their explanation (“we can only increase pay if hours change because that triggers a contract variation”) actually valid?
No. They can agree a contract variation with her at any time for any reason.
Could this be indirect discrimination if childcare prevents her increasing hours?
I don't think there is enough information here to say. She would need to consult a lawyer who specialises in employment law if she was considering going down this route.
Is a bonus clawback enforceable if she later reduces hours?
It depends. The bonus arrangement would have to include provision for a clawback from the outset and your friend would have to agree to that.
Is it appropriate for a legal firm to use discretionary sick pay as leverage?
I don't think there is enough information in your post to answer that question.
Should she ask for written confirmation that declining the promotion won’t affect her current job?
It wouldn't do any harm.
Does the fact she's been there under 2 years make any difference?
Unfortunately, it does. It means she does not have any protection against unfair dismissal unless her dismissal is unlawful discrimination or she is sacked for a reason that is classed as automatically unfair such as exercising her statutory rights.
Having said all that, I agree with the other posters that, on the information you have posted and assuming your friend has understood the email correctly, this doesn't sound like a good employer.