Hi Jimmy
If you are travelling from Scotland to N.I. you are not leaving the UK so there is absolutely no issue and no requirements. When you travel from N.I. to Southern Ireland there are no border controls so again absolutely No issue.
On the ferry the dog must stay in the van, (lucky dog, if we had the option we would also stay in the van), if he is anything like our mutt he will sleep throughout the crossing.
Enjoy Ireland.
Dee