re: How long should you stay in cities while backpacking?

Now's the time to do some reading. Get your hands on basic guidebooks for each city--while guidebooks aren't the key to a brilliant vacation, they will give you a good overview of the highlights. Your local public library probably has Fodor's, Lonely Planet, etc available through its e-book collections, so you can start your research without spending any money.

Bothering to read a few guidebooks will help you allocate your time. See what piques your interest....Roman ruins, WWII history, art, music, regional food specialties, hiking/outdoors....reading now makes for a better vacation later.

ETA: your entire list is cities--you might want to consider a few days in less urban environments. Smaller towns/villages can be picturesque and relaxing, esp after lots of concrete and people.