Is it legitimate for a state to ban the burqa?

The arguments on both sides are more complicated than presented here.

Should an employee be allowed to wear a cross at work?

In almost every case the answer should be “Yes”. There may be a pragmatic case for, say, banning loose chains that in certain workplaces may be dangerous; but it is difficult to see what right an employer has simply to ban the wearing of a cross as a religious symbol.

The question is not clear. Does the employee wish to wear it visible or not ? An employer has the right to demand that staff follow a uniform policy. If that policy says no jewellery then that includes representations of torture implements.

but it is difficult to see what right an employer has simply to ban the wearing of a cross as a religious symbol.

Really ? Your first commentator got it in one. Many employees are the public face of their employer. The employer might quite legitimately want that public face to be neutral from a religious and political perspective. That's one of the reasons employers want uniform policies. For example if I go to the bank to borrow money to set up my new business selling Islamic head scarves I might well feel uncomfortable if the person making the judgement on my loan is wearing an I Love Jesus badge.

Should a Catholic adoption agency be allowed to turn away gay prospective parents?

If the agency receives public funding, or performs a service on behalf of the state, then the answer is “No”. It would then be legitimate for the state to insist that the agency does not discriminate, despite Catholic views on homosexuality. If, however, it is a private agency – if it is simply performing a service for Catholic parents who subscribe to its views on homosexuality – then the answer should be “Yes”.

Adoption is always going to involve the state as it requires the state to agree to the adoption. So the second case should never occur.

Should gay marriage be legalised?

Yes. .... What the state should not do is to force religious bodies to accept or consecrate gay marriage.

Nope. You already covered this in the adoption agency example. The religious body is performing both a religious and a state marriage in one because it has been given a license by the state to perform marriages. With that license comes a responsibility to obey the state's rules about discrimination. Of course it can always refuse the license and perform only religious ceremonies that confer no legal marriage on the participants.

Michael