Not German, but I speak the language and I've been to Berlin a dozen times. My girlfriend studied there for 6 months. We love the city. So much in fact that we are going again for a week, leaving this saturday. I promise to keep my eyes open while I'm there and write down adresses if I come across anything interesting. It is a very bicycle-friendly city. What kind of bike are you looking for? Does it have to be a race bike? Based on your reply I'll do some Google-ing for you and post the links. My guess is there many options, you only have to choose one.
Things you need to know...a bicycle in German is a "fahrrad" and hiring is "vermietung" or "vermieten."
And concerning routes: might want to take a guided tour, for instance http://www.fahrradstation.com/fahrradst ... 6/cat/77/1
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They rent bikes as well!
Anyway, let me know and I will be in touch with more information.