1. Rome, 2. Paris, 3. Barcelona, 4. London, 5. Prague
If it's your first trip to Europe and there's no particular place you've always wanted to see, I'd go for Paris and Rome first of all. I haven't been to Barcelona but from what I know about it, that would be my third choice, with London perhaps the fourth (particularly for the art galleries and museums, not sure if you're into that kind of thing). Mind you none of these places are particularly cheap. I think the budget to live decently would be more like ~€100/night in Paris or Rome, or ~£100/night in the case of London, especially as you're going there in the peak season.
Copenhagen is nice and I'm sure you'll enjoy it; it might be a good opportunity to go there now if you can fly there cheaply but otherwise it's not a must-see place for your first trip. Still, even if you manage to get there without spending a lot of money, it's hardly a budget destination. Berlin you can skip; if you can only see one city in Germany, make it Munich. (Amsterdam and Dublin, I haven't been to.)
Outside of your list, I'd first of all suggest Prague; even if much of it has been converted to a tourist ghetto with hardly any local people venturing nearby, the old town is still picturesque, and the accommodation costs are much lower. I've also really enjoyed Brussels, and you can stay there comparatively cheaply too, for Western Europe. (Much) farther east, the old town in Riga is also very beautiful and staying there should not be costly at all. There are also many smaller cities that might be just as interesting for a short trip, one such place is Bruges. I've also always wanted to visit Lisbon, and I believe it's well worth it, and one more very decent city that seems to be very cheap to fly into is Stockholm.
Perhaps you will find something interesting among the suggestions above but on a general note: Europe is large, and you only have 9 days. The transport expenses will eat a huge portion of your budget and, perhaps more importantly, you'll just get tired from all the travel. I think the amount of time you have is more appropriate to visit 1-2 cities at most, and save the rest for your next trip.
Also, while you say the climate is not a concern, I wouldn't discount it entirely. Copenhagen will be very pleasant in the summer, while Rome might be hot (it was perfect when I went there in February). Just a footnote though, good luck with your trip!