1) If you were perfectly positioned and there was no air in that doughnut (you'd have to be in a space suit); then you would float free. That is, you would stay still; and, from your perspective, the station walls would appear to race by you at the speed the station was spinning. This would continue until microgravity forced you into contact with the inner wall (the side of the tube closest to the center of the station's center of mass).
2) No. friction plays no part, except that it might be used to impart tangential acceleration to you. If the station inside walls were smooth and frictionless (and that includes no air friction), the station would continue to slide by you, and you would remain weightless.
It is important to note that this is only a thought experiment. In reality, air friction (if nothing else) would QUICKLY accelerate you tangentially. That tangential acceleration would throw you against the outer space station walls, where the non-zero friction there would help the air friction in accelerating you tangentially even more. Once you gain any tangential velocity, it is ineveitable that you will contact the outer wall and gain radial acceleration, which you would percieve as weight.
To throw some numbers at this: A theoretical space station of radius 900 meters, doing a complete rotation every 60 seconds (in order to generate about 1g at the outer rim) would have an outer tube tangential velocity of about 90 m/s. The air in the tube would have the same tangential velocity. If you were instantly instantiated in this tube, weightless, with zero tangential velocity, you would experience an air blast somewhat similar to standing on the hood of a Ferrari, screaming down the Autobahn at 324 km/hr (roughly 190 mph).