FlyinMike - I'll try to add a little more to Milesilv's answer.
I'm not sure about taking the horses on your own as I am not a rider (although I wish I was just for this one stay). But we did see one guy out by himself riding along one of the paths near the tee box on the golf course. He was alone and I saw no guide in site. We saw another group of 3 woman riding through the parking lot near the main house but they didnt' seem to be getting any instruction or direction. You can email the Equestrian Manager and ask specific questions prior to your honeymoon.
www.mountjuliet.ie - great resource of information. I also emailed the reservations manager before we left to confirm our room and where we wanted to stay. They were always prompt with their replies.
We went to Waterford during our 3 night stay there. It took about 45 minutes on a Saturday morning. There was no traffic but I guess that there would be more on a weekday. I would also recommend going to the town of Kilkenny while your at the Mount Juliet, only 10 miles away, it is a great Irish town with lots of pubs, restaurants and things to see. I recommend the tour of the Kilkenny Castle, very neat.
Sorry I can't comment on the western half of Ireland as we jetted off to Edinburgh and London for the second half of our trip (hilton hotels report to follow).
Have a terrific honeymoon and I hope you enjoy the Mount Juliet as much as we did. One last thing, I found the Mount Juliet to VERY reasonable in terms of food, drink & activities unlike some other resort locations.