Is accretion disc caused by frame dragging? Is accretion disc around a Kerr black hole the result of its frame dragging?
 A: No, you get accretion disks around other bodies too, like stars during planetary system formation and especially in binaries when the companion star overflows the Roche lobe. They become particularly noticeable for dense objects, though.
The reason for an accretion disk is that gas and dust orbiting something will not spiral inward unless there is some dissipative interaction that allows it to shed energy and angular momentum. If we consider a random cloud around a dense object, interactions between constituent particles can make one particle go into a closer orbit while the other goes into a higher orbit (carrying off some angular momentum), plus release some energy that is radiated away. Conservation of overall angular momentum will also flatten the cloud into a disk orthogonal to the angular momentum vector as it self-interacts and loses energy. The result is an accretion disk where turbulence, friction, particle interactions and sometimes electromagnetic effects serve to dissipate energy that is radiated away. As mass spirals inwards potential energy becomes kinetic energy, and for dense objects the high velocities produce noticeable radiation.
Frame dragging matters only very close to a black hole. A more important limit is the innermost stable circular orbit (ISCO): as the disk matter passes it, the matter will plunge inward. This is the inner edge of the disk. One could image a counter-rotating disk where frame dragging makes the ISCO further out, but it is expected that the disk and the hole typically rotates in the same direction.
A: Accretion disk is not caused by frame dragging. Accretion disk is formed due to the gravitational attraction of matter surrounding a black hole. In general, black hole accretion is a transonic process. On the other hand, frame dragging is caused only around rotating black holes described by the Kerr metric.
For reference, you can look at the following papers:
Transonic Accretion and Winds Around Pseudo-Kerr Black Holes And Comparison with General Relativistic Solutions
Constraints on Transonic Black Hole Accretion
Rotating accretion flows: From Infinity to the Black Hole
Horizon-penetrating Transonic Accretion Discs Around Rotating Black Holes
