Do you have another equation that you can use to relate forces and acceleration that might be useful in this type of problem, in addition to ΣΓ = Iα?
I spent a little bit of time looking at Figure 2. One way you can attack this is just to assume direction for linear acceleration (and the corresponding angular acceleration) and direction of the resultant friction force and work from there. However, if you think about it a little bit, you can actually figure out the direction that the spool rolls and the direction of the friction force without having to write out anything. The way to do that is to assume motion in one direction and see if that makes sense. For example, let's assume the direction of linear acceleration for Figure 2 is to the left. Since force T is to the right, what direction would the friction force F have to be to cause acceleration to the left? Well, to the left, of course. And what would the magnitude of F have to be (relative to T) to produce linear acceleration to the left? Well, it would have to be greater than T. Next, look at the directions (and relative magnitudes) of the individual torques produced by forces T and F and see if it is possible for a CCW angular acceleration to result from that.