You can easily go to Belfast from Dublin by bus or train. Bus companies are Dublin Express, Aircoach or Translink. Depending on where you’re staying in Dublin, train could be convenient too. Last month in June we took the bus, but stayed in Belfast for a couple of nights. I think it’s definitely possible to do it as a day trip and see Belfast and take a black cab tour, but getting out to the Giants Causeway without a car might be difficult. I saw a public bus called the Causeway Rambler that you could take out to the Giants Causeway. I’m not sure how often they run or where they pick up from etc, so I can’t say how efficient that would be for a day trip with other activities planned. But could be an option.
As for Paddywagon Tours, we took a day trip with them on one of our Belfast days. It included Giants Causeway, Dunluce Castle and the Dark Hedges. I thought it was a fine tour. They showed up on time and the driver/guide was friendly and gave commentary between stops. The stops themselves were unguided, you were on your own. I think the tour you’re considering from Dublin would get you what you’re looking for without a lot of extra stress and planning.