It's worth taking into account that July is pretty much peak season for all of these destinations due to the summer holidays. Expect crowds, accommodation which is very expensive and/or booked full, etc. Rome might also be quite unpleasantly warm.
In Germany, if you want to stick to cities, Berlin is pretty much a must. So much history, and so much to see. However, I'd like to make a case for maybe trying for some smaller cities and seeing a bit of nature. Göttingen is my personal favorite unexpected gem to see for a day or two, but there's really not many mid-sized cities with which you can go wrong; you're quite spoiled for choice there. I'd steer clear of Frankfurt and Munich, though, and opinions on Hamburg are strongly divided, with a lot of people being very negative about it.
I'm personally not that much of a fan of Paris. It's too sprawling. Since I don't expect to talk you out of going, at least be very certain to pace yourself, and not try to cram too many different neighbourhoods in each day. Rome has a bit of the same problem (though with much more positive connotations for me); you'll need to really limit how much you try to put into each day there - there's just sooo much to see scattered around everywhere. I'd say 5+ days for each.
Amsterdam is much more walkable, and I guess you can limit your time there to 3 days and still get a good overview (though it pains me to say so, and really, longer is better). When there, be certain to head outside of the immediate city center, to the more leafy streets of the Jordaan (west from central station, beyond the Prisengracht), and the quieter canal sides along the southernmost stretches of the Prinsengracht and Keizersgracht. That's an entirely different side of Amsterdam. If you're around the museumplein area (I can highly recommend the Van Gogh museum), and want a retreat from the bustle of the city, head west from there to the Vondelpark. (Though note the general high season crowds modifier.)
I remember Innsbruck as more appealing than Salzburg, but haven't been to either in ages.
Actually, I was planning to stay in Munich for three days. Could you tell me what you didn't find appealing about Munich? At the moment, there is a possibility I could change my mind and choose Berlin.