What exerts the force of surface tension, and what does it act on? Let me start with the simple situation that I am familiar with.
This question might be kind of long.

In the situation shown in the above diagram, to keep the slider in equilibrium, we must exert a force F as shown towards the right.
My question is, do we need to exert this force to balance the force due to SURFACE TENSION towards the left direction?  Why will the liquid exert surface tension forces on a solid? I thought surface tension forces were between two liquid molecules on the surface
But, I came up with my own explanation:


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*The molecules in contact with the slider experience adhesive forces towards the right direction.

*The molecules just to the left of these molecules exert forces of surface tension towards the left direction, on these molecules in contact. 

*Since the molecules in contact with the slider are also in equilibrium, the adhesive forces are equal to the surface tension forces.

*By Newton's third law, since the slider pulls the molecules in contact with it due to adhesive forces, the molecules pull the slider with an equal and opposite force, with I proved in point 3 to be equal in magnitude to the surface tension forces.

*This, I have concluded that the force F needed would be 2Sl, to maintain equilibrium of the slider.
Is this reasoning correct?
Consider a thin capillary is dipped in water. The water rises up. I have learnt that the adhesive force acting on the meniscus due to the walls of the capillary is always NORMAL TO THE SURFACE OF THE CAPILLARY. But, the surface tension forces acting on molecules just around those molecules in contact with the capillary walls are not in the same or opposite direction as the adhesive forces experienced by the molecules in contact with the walls of the capillary. They are tangential to the surface of the liquid. So my reasoning fails to account for the force of attraction between the glass and the water.
To summarize, I have two main doubts:


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*Is the force due to surface tension cohesive only?(does it only act between molecules of the same liquid at the surface?)

*The surface tension forces acting on the edges of the capillary help to support the weight of the liquid column. This implies that the adhesive forces between the glass and liquid have a vertical component. But I have learnt that the adhesive forces are horizontal. So, who exerts this force on the water?
 A: Surface tension is a quite confusing subject, especially viewed from a purely mechanical point of view. It appears whenever you have an interface between a condensed phase say $A$ and another immiscible fluid phase $B$. 
Thus the first thing to note is that surface tension has always to do with an interface. The surface tension coefficient often denoted $\gamma_{A,B}$ will tell how "costly" it is, in term of energy, for such an interface to exist.
Now the reason why it is costly to have such an interface is ultimately due to the effective adhesion forces between the molecules in each phase. To simplify a bit, there are two principles at play:


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*(1) In a quite good approximation, molecules interact with van der Waals (vdW) interactions which are always attractive (in vacuum). Furthermore, the vdW forces are the strongest with molecules of the same kind.

*(2) In a dense phase of certain molecules, the cohesive energy density is higher than for the same molecules in a more dilute phase.
These two rules have two implications:


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*If phases $A$ and $B$ comprises the same molecules but have very different densities (e.g. liquid water/water vapour interface), then by the rule (2) there is a big loss in cohesive energy density for each piece of interface created between the two phases. From a mechanical point of view, it is fine to say that molecules in the liquid phase are simply pulled stronger towards the liquid phase than the gas phase.

*If phases $A$ and $B$ are two condensed phases comprising different molecules, then by the rule (1), it is also costly to generate an interface between $A$ and $B$.
This leads to the property that the surface tension coefficient $\gamma_{AB}$ is always positive.
Now, in most real cases, multiple interfaces are involved at the same time. Most of the time three interfaces. This is the case for the meniscus you mention but also for the insects walking on water.
To discuss the insect example, one needs to guess whether its legs are wetting or not. If they were, then it is likely that it could not walk on water as it would be preferable for it to actually sink in water. It must have quite a lot of short straight hairs on the legs to induce a hydrophobic effect effectively "repelling" water and inducing only a single contact point with water and then one only needs to care about the deformation of the water/air interface.
Now, regarding the direction of the force, one needs to discriminate two things:


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*the total interaction between a phase $A$ and a phase $B$

*the surface tension interaction between phase $A$ and $B$
While the former accounts for all possible forces between the phases, the latter is only concerned with the shape of an interface and acts by definition tangentially to the interface.
For example, in the first example you mention, this is a mixture of both:
First, the liquid wets the rope which more or less implies a strong adhesion with it, second the liquid exerts a tension related to the $\gamma_{air/soap}$ interface which acts along the interface air/soap but perpendicular to the interface rope/soap; that's mainly because we consider ourselves in a case of ultra-ideal wetting. Thus what it says is that Nature prefers gaining a bit of energy by extending the interface air/soap a bit rather than gaining a much bigger amount of energy by detaching the rope or whatever object you might use from the soap film.
Try the same experiment with a tube made of GoreTex, I am not sure you would get the same outcome.
