"Does it have the right to reject for employment someone who is "living in sin"? Yes the interviewers could give a fudged reason for rejection but could they actually say "we rejected you because you are in a non-marital sexual relationship" in the same way that they want to be able to say "we rejected you because you are homosexual."
I don't know. If one of the job requirements was that the candidate be someone who attended Mass regularly and was a practising Catholic, then they probably could reject explicitly for that reason as - like it or not - both those things are incompatible with Church teaching.
"Also once employed and having it discovered that they were "living in sin" would they be able to be sacked?"
I don't know, honestly, what the position in law would be but I suspect that there would be little comeback for the employee (so long as proper termination procedures were followed and so long as the requirement to be living a Catholic life was made clear at the time the employment started).
In the case you mention, lets take the example of someone working for a Catholic marriage guidance charity, where their job involves helping and advising those getting married or those whose marriages are in difficulty. Any advice/counselling would be given from a Catholic POV and would revolve around church teaching. If it then came to light that the advisor/counsellor was telling couple one thing but practising another (eg: contraception - though how that would come to light, I don't know! - adultery, living in sin etc), then there would be an inherent hypocrisy involved - something the Church would like to avoid, I'm sure.
All of this is a dilemma for all religions - society has changed dramatically and in many cases for the better but religions are much slower to change - if, indeed, they are going to change at all.
As an aside, I'm a pracising Catholic and, like many others, I sometimes struggle with my faith. The key thing I try to do is to remember that it is not for me to judge other people. If someone I knew was flouting Church rules by, say, having extramarital sex but was till taking communion I wouldn't say anything and I would try not to think badly of them - how do I know that I wouldn't act the same if I was in their shoes? The church teaches that, through confession, the individual reconciles themselves to God. There are very clear rules laid down by the Church to guide our behaviour in all aspects of our life and if we choose not to follow all of the rules, then it is up to us - as individuals - to reconcile that with our consciences. I'm not always a good person (who is?) but I try to be and I try to remember that Christ had compassion for the weak and the poor and the outcast - and that is the example I should follow.
Saying that, I would find it uncomfortable if someone was representing the Church in some capacity (as a charity worker or as a teacher etc) and was saying one thing while doing another but it would be up to the Church to take action if it wished, not up to me. Does that make sense?