In Ireland (where Dublin and Castle Leslie are located), you will use Euro. If you plan to go to Northern Ireland (where Belfast and Ciaran's hometown are located), you will use pounds.
If you do not plan on renting a car, you can get to Monaghan by bus from both Belfast or Dublin. However please note you will still have to take a cab from the bus stop once dropped off in Monaghan. Please email kpcm713@gmail.com for all bus information and schedules! (I made a really great google doc but can't figure out how to put it on this website. So just email me and I'll send to you!)
There is no border patrol or anything when crossing to/from Northern Ireland and Ireland. It is similar to crossing state lines in the US. The only difference is that the road signs will change. Ireland speed limit signs are in Kilometers and Northern Ireland's are in MPH. Rental cars will display both.
Yes, you can rent a car with your US driver's license. Please note that rental car places will have many manual car options, so make sure to rent an automatic if that is what you are used to. Driving on the left side of the road was not as difficult as I thought it would be. You get the hang of it after a couple of minutes. One thing to remember, Ireland has a lot of round-abouts so just make sure you enter left and not right!