I spent a year of my degree scheme working in Erlangen, located in the traditional and often stubborn Franconia. I turned up with limited German, and had intended to learn the language in situ. However, due to my employment there being in an international office and language learning becoming a more expensive mission than anticipated, I ended up not leaving fluent, as I had initially anticipated.
To relate this to your questions:
I was wondering: How true is that?
English has Germanic roots. You'll be surprised how much vocabulary you'll be able to translate through guesswork. A guidebook for reference along the way may 'save' you from ever truley needing to learn the language while still surviving (it worked great for me!)
Does really everyone know English?
Big cities and what was West Germany has a very high percentage of English speakers. In the old East Germany and 'traditional' states you may find that the older generations (i.e. those 40+) do not speak English well or at all, especially away from the big towns, but anyone below 30 or so has had to complete compulsory English at school so will almost certainly have stronger English than you will German.
How much of the population is comfortable speaking it?
Especially if you have some basic German, politely try and say that you'd be able to communicate better in English, I never found anyone that wouldn't give a bit of English a go. Occasionally their English was poor, but they were 1 in 100 cases. Anyone in medicine or engineering will almost certainly speak English very well.
Are people happy speaking English casually, or do most people still
prefer to converse in German?
I began a system at parties where I would be spoken to in German and respond in English. People will of course prefer to speak in their own language, but if you are a foreigner and making friends, I found almost all I met to be very accommodating and supportive folk, willing to communicate in the most mutually beneficial language.
Be polite and learn enough German to buy dinner and ask for directions, but for almost all other responsibilities I believe that it is true that many Germans would find you struggling with their language endearing but ultimately a waste of time when they can and will speak in English with you.