Quite a bit of 'capital city syndrome' in this thread. Anyroad, in answer to the original post.
Amsterdam:
From my experience prices are similar to the rest of Europe aslong as you're not an idiot about it e.g. chains are the same as everywhere else, some places however are aimed alot more at tourists so you're looking at a few euros extra here and there. Overall though not too bad at all.
Everyone I met spoke perfect English, including homeless people which was a bit unnerving
Hash cakes were available in a few cafes I noticed and were about what you'd pay for normal cake in a coffee shop.
Can't comment on the week night life as I was there Friday - Monday.
Had no experience with crime. Few shady looking guys in some of the clubs but nothing came of this. Police I encountered were helpful with directions and questions.
As far as the sex district, yeah it's fairly obvious and fairly seedy but is pretty seperate; a city within a city if you will.
Rome
Been a while since I went but it's a stunning city although fairly expensive both to stay and eat out unless you know decent places off the beaten track. Most staff spoke good English, the public were about 50/50 but this may of just been my experience. Worth checking out but if you're on a tight budget then maybe leave it for another time.
Prague
Heading there myself in a couple of weeks but haven't been before so can't comment I'm afraid.
London
One of my favourite cities to visit; feels like four or five cities combined, each with their own personalities. Can be quite expensive in some places (hotels on average are around 10-15% more just because of the location) but overall no worse than anywhere else. Pound is stronger than the Euro at the moment, but it's at it's lowest for quite some time so you may want to take advantage of it.
Literally no end of things to do so I suggest you plan what you want to see so you fit it in. unless you're like me and just like wandering around. Most museums are funded by the DCMS so are free entry, some of the city attractions however can be a bit pricey.
London does have some gang issues (which isn't suprising as a fifth of the population live there) but crime shouldn't be an issue as I imagine you'll be staying to the tourist areas and not heading into the slum parts. Aside from not having your wallet on display on the tube you'll be fine. Note that Londoners can be a bit rude on the underground so make sure you stick to the correct side of the escalators.
Berlin
I liked it but I was too young to appreciate things like nightlife so I can't really comment. It's a pretty beautiful place to walk around though and from what I remember prices were suprisingly good for a capital city (note this was about 7 years ago though.)
Cologne
Never been I'm afraid.
On your budget I'd recommend Prague, Amsterdam and London, in that order; Prague for the scenary and the food, Amsterdam for the cafes and night life and London for the museums and the city itself. Save London for last as it's very easy to go over budget there and doing it too early may cut your trip short.