So you have 12 days to play with - start by being realistic (and honest). If you're flying in from someplace far away, your arrival day (and your departure day) are pretty much a write-off - you can't count them as usable days.
Also, at that time of the year, days (daylight) is very short, so it will limit how much you can do each day (besides having limited daylight, some things close early - especially around major holidays).
With 12 usable days, I think you need to reduce your itinerary. The obvious outlier is Rome. Save that for another trip when you can do it (and more of Italy) justice. With 12 days, in the dead of winter, for your first trip, Dublin, London and Paris will be plenty. Many people make the mistake of trying to move too fast - we all (usually) learn that slowing down gets you a better trip.
Have you booked your flights yet? You can save yourself both money and (more critically) also save time by flying open jaws: fly in to one city, and out of another (so no time and money wasted circling back to your arrival point - this effectively "buys" you 1-2 more usable days - you get a longer trip just by doing this). If it were me, I'd fly in to Dublin, do 3-4 days there, probably fly to London and do 3-4 days there, then take the train to Paris, do 3-4 days there, fly home directly from Paris. Skip Italy this time, it'll still be there next time you go.
Other than the advice above, if you haven't already seen them all too many times, you should watch the Rick Steves videos on Dublin, London and Paris. Also look for his "travel skills" video, it has lots of useful info. Get his guidebooks for each location and start reading them, too.
Hope that helps.