What I mean is that the European directive which let you apply for the residence card are the same in Ireland and in the UK (or any other EU country) - so they will not rely on the UK's decision but they will assess the application based on the same criteria.
It is strange your UK application was rejected just on the basis of not finding you at home though ... under the directive you should definitely have been given an opportunity to clarify the situation.
In any case you can come to Ireland directly without a visa (showing you marriage certificate at the border and the officer will decide on the spot whether to let you in), or the other option is to get a visa (free of charge and fairly straight forward) so that the decision is made beforehand. My advice would be to apply for the visa so that you are sure they will not stop you at the border due to the UK refusal (based on what you said they shouldn't, but better be safe).
Once you are in Ireland you can apply for a residence card using this form (your spouse needs to find employment or become a full time student first):
http://www.inis.gov.ie/en/INIS/Form%20E ... %20EU1.pdf
And also you don't need to get married again ine Ireland - a UK marriage certificate is perfectly valid here.