Your question consists of three parts so let's answer them one by one.
1) Surface tension between liquid-solid and air-solid
why isn't all strange surface tension phenomenon seen at those surfaces?
Just a short note upfront: I don't know what you mean by 'strange surface tension phenomenon' so I will explain in general how surface tension is also present for the liquid-solid and the air-solid interface.
You are absolutely correct in saying that these interactions also exist at other interfaces, so also at the interface of the liquid and the container (solid) and at the interface of the air and the container. The main reason why this is not easy to see is that the solid is extremely rigid as compared to the liquid and gas and therefore doesn't significantly deform because of surface tension acting on it.
You can, however, see a signature of the surface tensions of gas-solid and liquid-solid in the case of a sessile drop. What happens there is that the droplet while adjust it's shape to a certain contact angle ($\theta$). According to the balance of there 3 surface tensions. This can be visualized as follows:
If you balance the surface tensions you get an equation which is called Young's equation: $\gamma_{sg}=\gamma_{sl}+\gamma_{lg}\cos\theta$. If, for example, the surface tension between the liquid and the solid is extremely high, this will result in a contact angle that is close to 0.
2) Direction of surface tension
how can surface tension be described as perpendicular forces to an imaginary line at the surface, isn't it supposed to act in all direction?
This is actually not true. Surface tension is a force parallel to the surface. The reason comes from the molecular origin of surface tension. Let's explain it in terms of the liquid-air interface. Liquid molecules prefer to have a certain amount of interaction with their surrounding liquid molecules. If a molecule is located at the air-liquid interface, it doesn't have sufficient interaction with the air molecules. To 'solve' this, the molecules at the surface will pull more strongly on the neighbors that are also on the surface, thus exerting a force parallel to the surface.
3) Capillarity
how can we assume the meniscus formed will be spherical and not defined by some other function?
I assume you meant to write the equation for capillary action: $h = \frac{2\gamma \cos \theta}{\rho g r}$. The assumption there is indeed that the meniscus is spherical. This is necessary to simplify the mathematics and be able to solve the liquid height in a capillary tube analytically.
You can estimate when it is still a reasonable assumption by calculating the capillary length: $\lambda_c=\sqrt{\frac{\gamma}{\rho g}}$ and comparing this to the radius of the tube. If the radius of the tube is below this length then surface tension is 'sufficiently dominant' to make sure that the meniscus is a spherical cap. You can see this capillary length in the equation for capillary action:
$$h = \frac{2\gamma \cos \theta}{\rho g r}= \frac{2 \lambda_c^2 \cos \theta}{r} $$
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
Surface tension is rightly named: it is a tension in the surface that separates a liquid and another fluid, air in your case. This tension is at the origin of forces: e.g, if the surface is curved, the force tends to straighten it (as when you exert a tension on a piece of string: it will straighten). In a small spherical bubble which is far from the walls of the container and from the surface, surface tension acts all over the surface, exerting a force toward the center of the bubble: this force is resisted by an opposing pressure in the bubble. This is known as Laplace law.
If the bubble is at the bottom, then surface tension is also at the origin of a force at the line of contact of the water-air interface and the bottom of the container. This force is proportional to the value of the surface tension (a constant that only depends on the fluids you have and temperature) times the length of that line.
If the bubble is small enough, this force can resist the buoyancy force. Imagine now that you inflate the bubble, the volume and thereofre buoyancy will grow faster than the perimeter of the contact, and at some point the buoyancy force will be able to detach the bubble.
(Note that in order to have a complete picture, one should also consider the shape evolution of the bubble, which is governed by other surface energies: the solid-water energy and solid-gas energy, as the angle of contact governs the proportionality mentioned above)