Is cause of this equilibrium - tension or friction? This is a very odd problem which I have.
As you can see in the diagram, the string is taut. So, if there was no friction you could clearly say that tension is balancing it or if there was no tension then friction is balancing it.
But in this case where you have both tension and friction which would actually be the cause of the equilibrium ?
Or if we put it in another way, what should be the tension in the string ? 0 or greater?
 A: If you draw the free body diagram, you will get $$T = mg \sin \theta - \mu mg \cos \theta = mg (\sin \theta - \mu \cos \theta) $$ So, $T>0$ since $\theta$ is non-zero, $\mu$ is non-zero and $mg$ is also non-zero.
It is due to the equilibrium that we get to write the above equation. The 2 forces, namely tension and friction are in same direction as the gravitational component of the mass' weight is in the opposite direction.
If $T = 0$ then the mass would fall owing to the gravitational component of the mass' weight provided $$\mu \not = \tan \theta$$
If $\mu = \tan \theta$, then the mass would rest on the wedge even when the string has been detached, i.e. tension $T = 0$.
A: As stated, mathematically, any fraction of tension and friction is possible. Maybe this will help you to wrap your mind around this perhaps seemingly paradoxical situation: This math, and the mechanics it describes, is and can only ever be an approximation to the real world. Imagine you had actually balanced things so perfectly that the mass is supported by friction alone and the mass-less string only looks taunt but has zero tension. Then any ever so slight vibration would cause the mass to slide---if your idealized, mass-less and infinitely stiff string would not assume any tension necessary to prevent it from being stretched further even just the tiniest fraction of a micrometer.
In real life, unless you did not choose an incline where the mass was really about to slide even just from breathing on it, such a microscopic slide would occur and stretch the string to the point where it does stop further motion. The exact ratio of tensile and frictional force is then determined by possibly minuscule details of such a micro slide, the string's elasticity, the non-static coefficient of friction, etc.
A: You have one information (total tangential force) for two unknowns: there is no answer to this question as-is. Any combination of those forces will yield the expected behaviour.
If you want to know the actual tension/friction ratio, you need a more detailed description, maybe considering the rope as elastic.
A: Here's a gedanken experiment to show that there is no way to know the tension. Imagine that you have pulled the block slowly up to a certain position. The tension will be the sum of the gravitational force in the string direction plus the friction force, whether it be just a simple dynamic friction force or the static friction. You can slowly release tension on the string and the block will not move, but the tension on the string will decrease until the gravitational effect overcomes the friction. Therefore, it cannot be calculated.
