Too many questions for one post, but I'll try a few. It's very easy to fly from Dublin to the UK or most European countries. Have a look online at who flies into and out of Dublin and where to.

http://www.dublinairport.com/gns/flight-information/destinations-airlines.aspx

Only you can say where you need to see. There are lots of places that it would be nice to see, but budget and time knocks a lot of them out. You say that "I" will be flying, but then should "we" rent a car. It all depends on how many of you there will be and how important independence of travel is to you, as well as cost.

Have look at http://www.viamichelin.co.uk/ to work out your mileage and potential cost for fuel.

If you plan to do that loop around Ireland, you won't have time to go to Liverpool or Italy or anywhere else. That's actually a lot to put into a week, just with Ireland.

"Are hostels better" - I can't speak to Dublin; as a bigger city, it's more expensive. But when I visited Kerry and Clare a few years ago, there wasn't much difference in costs between hostels and bed & breakfasts, and the B&Bs are much more comfortable (and obviously you get breakfast!).

If you plan to go around the countryside, a car will give you much more flexibility.

My advice is to look for trip reports and threads here about Ireland - that will give you a good idea of what others have seen in the time you have, and how long it takes to get from place to place. Things move more slowly there than you might realize!