The mechanism is explained, e.g., in W. Johnson, Int. J. Impact Engng, Vol.21, Nos 1-2, pp. 15-24 and 25-34. 1998.
The following main assumptions are used to derive the approximate Birkhoff formula for the critical ricochet angle for a spherical projectile:
(i) The pressure $p$ on a spherical surface element along its outward drawn normal is
$\rho u^2/2$; u is the forward speed of the sphere resolved along the normal.
(ii) The pressure applies only to those parts of the sphere which are immersed below the
undisturbed surface of the water. The effect of the splash on the sphere is considered
not to contribute any pressure.
Thus, I believe, surface tension is negligible.
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
The real issue is that the cup wasn't really full so that adding anything more would make it spill. You can clearly see the the level slowly growing above the top of the cup, as would be expected due to surface tension. Eventually another coin finally exceeded the limit, and a little water spilled. There is really nothing extraordinary going on here.
They could have just as well added some more water as individual drops and gotten the same effect. You say that the cup was filled so that another drop of water would cause it to spill, but that was never stated nor demonstrated in the video.
Once difference between adding a water drop and a coin is that, if done carefully, the coin will cause less of a wave. In fact the coin that caused the spill seemed to be added deliberately to cause a wave, like the person was tired of adding coins and wanted to see the spill already. A coin can be inserted into the water edge-on, and cause a small wave when doing so. A water drop will cause more of a wave because the surface tension of the water in the glass and that of the drop merge when they touch, which causes a sort of snap action that cause a wave. This wave will more likely stress the miniscus at the edge to the breaking point than the tiny rise in water level due to the drop alone.