Should the EU move towards a more integrated legal system with uniform civil and criminal laws across all member states?

Yes

No, national jurisdictions should retain a degree of sovereignty in terms of laws, however greater uniformity on transnational legal issues should be introduced, e.g. trafficking crimes; anti-money laundering, etc.

Yes but only if it was voluntary and I agreed with the all the laws

we should also have our own laws based on the unique cultural and social challenges we face as Irish people

In certain laws yes. but this question is too broad as it would take reviews of individual laws to make it solid.

yes but each member state should maintain their own systems and jurisdiction

I think this has potential. For example, if someone committed a crime here in Ireland and they went to France, they should be able to arrest him and sent him back here. Other than that I'm not one hundred percent sure.

No, but a shared criminal database across member states should be implemented

Yes but only to maintain a standard outside corruption influences.

No, different societies and cultures will have different views on civil laws

Within the common travel area the basis of all laws should be regarded in a shared cultural or linguistic basis. The basis of western morality is from our shared religion/morality and thus most of the laws translate but some don't. A consensus on the arguably archaic and cultural differences shouldn't necessarily be dismissed but should be given leeway to those who are not native but repeat offenders should be penalised.