It is known that molecules at the surface are strongly attached to each other (more attraction less repulsion) than those within the bulk attraction and repulsion are balanced).
I don't think that is an accurate description. Molecules in the bulk are maximally surrounded by neighbors and experience the most attractive force. The forces on a bulk molecule are balanced directionally. Molecules at the surface have fewer neighbors and experience less attractive forces. The forces on a surface molecule are not balanced, there is net attraction in the inward direction with respect to the fluid.
why evaporation occurs towards molecules of the surface (the stronger attachment than those of the bulk)
Again, the surface molecules experience less intermolecular attractive forces, not more. This is why liquids are in the lowest energy state when surface area is minimized. Droplets are spherical absent external forces.
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:
(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:
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:
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.
Best Answer
It is not simply the water-air surface tension that allows the insect to walk on water. It is the combination of the legs not being wetted and the surface tension. The legs of water striders are hydrophobic.
Water molecules are strongly attracted to one another. This is due to "hydrogen bonding": a proton in water is shared between two oxygen atoms of two water molecules. Considering only water and air, minimizing the interface surface area is the lowest energy state, because it allows for maximum interaction between water molecules. If the water molecules were attracted to the molecules of the insect legs and wetted them, the legs would sink into the liquid. However, in the context of the legs not being wetted, the attractive forces of the water molecules result in a net upward force on the legs of the insect as the legs deform the surface.