$F_f = \mu m g \cos(\theta)$
This is the maximum value of friction on an object, not necessarily the actual friction. If the ramp is horizontal, you'll calculate a large value, when the actual friction will be zero.
For the static case, the actual friction will be no larger than the component of gravity parallel to the ramp that is pulling it to the right.
I'd say that on surfaces with large friction, the cube would tip over sooner.
You didn't do the analysis this way, but in the static case, the object will never tip over unless the center of mass is to the right of the corner. In the case of a cube, that means a 45 degree angle. Raising the coefficient of friction doesn't make it tip over sooner. It just keeps it from sliding away before it tips.
If you're considering a sliding case (the block is already moving to the right with "sufficient speed"), then your analysis is already correct and the angle is irrelevant.
And a final problem (if you're trying to match what you see with actual blocks and imperfect ramps) is that the modeled friction force isn't always realistic once edges appear. If that lower corner of the cube "catches" a bump on the ramp, then the forces that appear can be much larger than what you'd calculate by the friction equation. So sharp edges are going to allow it to tumble in cases where the simple calculation says it should not.
Best Answer
The answer seems rather obvious to me : That an object "slips off the surface" means that there is no longer any contact between them. When there is no contact, there is no contact force. This applies to the normal force and friction, which are both contact forces.