1. Will trains and buses run after 10 PM?
Trains should run until about 12:30. As long as planes are landing, buses are running.
2. I hear there is a limousine bus that goes to Hiroshima Bus Terminal. How much does this cost?
I am not familiar with that. I am not seeing it on the KIX site.
Kansai International Airport | Access Information | Bus
4. If I take a train, it stops at JR Hiroshima station. The hotel website says it is 5 minutes away by taxi. I don't have a car seat for my baby, won't that be a problem?
Legally? No. Just hold your baby for the ride.
5. By the time I get to Hiroshima, it may be almost midnight. Will I have a problem getting transportation?
No, taxis run all night and will be readily available from places like train stations and bus depots. I am guessing the Hiroshima bus terminal is very close to if not connected to the train station.