I live about an hour from Edinburgh and of your 3 options I've been to bits of them all.
Option 1.
Dublin is ok for a few days but if you want architecture/scenery there are better options. Yes Temple Bar is a good night out but it (and Ireland as a whole) is very very expensive. Edinburgh is a nice city for architecture but don't forget about Glasgow. The West End of Glasgow has some really nice old buildings and a mix of modern and classical museums (Riverside is very modern from an architectural POV then you have the Kelvingrove thats very old and a stunning building). Glasgow and Edinburgh are only an hour or so apart by train and if you want scenery you can take a train north through the Highlands to Fort William which will take you 3 or 4 hours. As is normal with Scotland if the weather is nice its stunning but if its wet its all a bit dull.
Option 2
I've only been to Amsterdam but it is very vibrant and has some great museums and architecture. Obviously it also has the girls and the drugs but that ain't my thing so can't really comment on that. From what I hear Copenhagen is nice but also expensive.
Option 3
I've been to Prague and Budapest. Again two great architectural cities. I preferred Budapest because I felt Prague was ruined a bit by being too touristy and too many stag and hen parties due to it being so cheap. When I went I started in Venice then onto Ljubljana, Zagreb, Budapest, Krakow, Prague. Budapest was by far my favourite city. I can't really pinpoint why it just had a great atmosphere, cheap, really nice buildings and bridges. There and Ljubljana are two places I'd definitely return to.
With regards trains and things we
interrailled and it was the cheapest and easiest way to get around. The great thing about the train is you get a chance to see the scenery.
Whatever you decide have a great trip!!
Neil