The general rule of the most significant sector applies on all legs

You can check in up to 23kg at SEA and this will be carried to FCO via DUB without need for recheck, assuming this is all on the same ticket and a same day layover in Dublin
You are entitled to a 10kg carry on bag from SEA to DUB and from DUB to FCO


If you flew DUB FCO on a separate ticket, then you would have to pay for checked luggage and also check in the carry on unless you qualified to carry it onboard per the rules

• AerClub - Silver, Platinum and Concierge AerClub members (Members only, not travelling companions)
• AerClub Reward Flight - on all AerClub Reward Flight bookings
• Fare Types - All Plus, Advantage, AerSpace and Economy fares purchased via British Airways (Not included in Saver fares)
• Infants - Accompanying adult travelling with an infant
• Connections - Customers connecting to/from an Aer Lingus transatlantic flight or a connecting flight with an interline partner
• Aer Credit Card - Customers who are travelling using their Travel Reward flights to Europe