Hair dryer: Unless you already have one with a voltage switch, don't worry about bringing it... lest you purchase a voltage converter. Back when I had long hair, I simply bought a compact 220V model in England for £10 and always just took that one to Europe with me.
Shannon to Dublin: Your best bet is probably to catch an airport bus into Limerick and then take the train to Dublin. Alternately, you can go to Galway and then head to Dublin.
Sleeping bags: While many hostels won't allow you to use them, you can certainly have them with you. You just aren't supposed to use them on the beds, as it's one way for bed bugs to get around.
Money: $2700 should be enough for the two of you. The exchange rate is at an alltime low right now, though, so budget wisely and eat on the cheap!
There are a zillion various posts about things to do in the cities you're planning on hitting, so check out the Western Europe section to see what's cool and what people like....
In Dublin, since you're beer drinkers, you should definitely check out the Guinness Storehouse (brewery tour). Just don't buy anything at the giftshop, as it's overpriced, and you can get better deals on stuff online or in the Duty Free.
The Heineken Brewery Tour in Amsterdam is very worthwhile. You'll never get a fresher Heiny and it's relatively cheap, especially compared to Dublin.
London's another expensive one (damn the exchange rate!), but don't deny yourself the pleasure of having a few pints of cask ("real") ale at just about any pub. If you're staying in the Bayswater area, go to the Redan pub on Queensway for 2-for-1 meals and good cask.
Sorry, don't have any tips for Austria or Italy