1 week trip-what do you recommend?

Hello My family and I (4 of us incl. 2 young kids < 10) are planning a 2 week trip to Ireland and the UK (mainly London). Our plans are to spend a week each in both places. Some questions-

1. What would you recommend as absolute must-sees on a weekly trip?
2. Any kid friendly activities/places that can keep the parents awake, unique to Ireland?
3. Would you fly to London? What is the most cost effective way to get there?
4. Would we need to rent a car to get around or depend on public transit? How is parking around places to visit?

Appreciate any answers, tips and feedback shared here. Thank you