Yeah, I don't understand why you would have/need a residence permit with an F visa.
As said above, you have to register with the police station so they know where you are staying but that's something different.
When an alien wishes to lodge at the home of a Chinese resident in urban areas,the host or the lodger shall,with in 24 hours of the lodger's arrival,report to the local public security organ with the lodger's passport and certificate as well as the host's
residence booklet and fill in registration forms of temporary accommodation.In rural areas,the host or lodger shall report to the local police station or residence registration office within 72 hours of the lodger's arrival.
Actually, they are not strict about that 24 hour time period. Your landlord doesn't have to drop everything at a moment's notice for you to register.
I believe most F visas are for three months. I believe most F visas are single-entry.
You are not supposed to draw a salary from a Chinese company on an F visa. People do this ALL the time and don't get in trouble. However, if you make an enemy the police might get you on this.
There was a recent incident of a stalker on thebeijinger.com who was deported. He was working on an F visa and I believe that gave the PSB officer an easy way to get rid of him.
A Z visa is a good option if you are living long term in Beijing. It's not too expensive (RMB 2000 - 3000 for a years extention) and you have peace of mind. Even if you obtained the visa within lets say a "legal grey area", I personally believe that won't
matter. In my experience of China, a certificate obtained through bribery is equally as valid as one that was obtained with all the correct documentation. The way that Chinese officials work is that if you've got the correct paperwork they process you and
couldn't give a flying kick how you got it. If anything is out of place or unusual they will take the opportunity to screw with you.