London for 2 days, Paris for 2 days, Dublin for a day-and-a-half, Milan for 3 days, Venice for 1 day, plus travel in between? Honestly, I think this is a LOT, and will be a LOT of schlepping!
Two days in either London or Paris is not NEARLY enough time. And yes, you could make a "flying visit," but with travel factored in, you will be really shortchanging yourselves.
At a minimum I would assume I was going to go back many times over the years, and just delete London from my itinerary. Venice is only about a 3 hour drive from Milan, so... there's an argument to made for cramming it in there - but then there's Dublin. Maybe delete Dublin as well, and up your time in Italy? (Or factor that extra day in as travel time to Rome, which is more than a 5 hour drive south from Venice...)
Honestly, I think you will have a much better time - and be able to savor and appreciate a lot more - if you trim down the number of cities, then put things in an order that shortens travel in between.
As for the Moulin Rouge... yipes. I think there will be quite a few comments about why that might be considered a tourist "trap" and a bit of a waste of money. If you have your heart set on it, okay, but I can think of a lot of other places I would go to first in Paris - and many other places I'd spend my food dollar. I'd suggest taking a romantic stroll through the back streets of Montmartre, and making sure you see the Moulin de la Galette, which is very cute. You could then see the Moulin Rouge from the outside just before you hopped on the Metro and went somewhere else. :-)
Just my two cents'!
And an "oops!" I just noticed that you are flying out of Dublin (and into Dublin?) b/c of airfare. That does change things. Maybe get rid of Paris rather than London, this time.
Others will of course have SUPERB feedback!