Just have to add a comment on some things.

It is unfortunate that they refer to the 26 dimensions...

> [...] particles as tiny vibrating loops of string that exist in somewhere between 10 and 26 dimensions.

... of what is known as "Bosonic string theory". It is called bosonic because it only has bosons (e.g. photons) and no fermions (e.g. electrons). This is obviously not a realistic theory because of that reason, and it also suffers other serious problems. But, this was the first formulation of string theory, not meant as a fundamental theory of gravity even, and if you do open some of the famous text books in string theory, you do find that it starts with the bosonic string theory. This is because it is a more gentle introduction.

The "between 10 and 26" comment is also a bit unfortunate. String theory is ten dimensional (space-time, meaning I include time in there). A lot of physics is formulated in terms of perturbation theory, meaning you have that the full result is expressed as an infinite sum of smaller and smaller terms, and you can truncated this infinite series and get a approximate result. This holds if the parameter you are expanding indeed is ever-smaller, which it isn't necessarily. One of those parameters (string theory has two of them built in) is the string coupling "g_s". If you start taking this parameter large than one, so the perturbation breaks down, string theory (type IIA in particular) grows an extra dimension into a theory known as M-theory. Note that this theory has no strings, it only has other fundamental objects. Similarly, there is an F-theory that is in some sense 12D, which also describes non-perturbative physics.

So, if physics in our universe is described by this non-perturbative physics, then sure, it's 11D or so, but we do not know which parameter regime of string theory our universe is in ( yet ;) ). But it is not a choice willy-nilly.

Then regarding effective theories against fundamental ones. Effective theories, or models rather, are things like: the inflationary model, cosmological constant to explain dark energy, standard model, minimally-supersymmetric standard model, F(R) gravity, DBI gravity, and so on. The problem is that there are too many of them. Claudia de Rham had a talk a month or so back in which she said something along the lines of (this is how I remember it) "We are quite good at excluding effective gravitational models, but we are however better at constructing new ones.". We need some deeper understanding of what is allowed when it comes to model building, and even theory building. But the point is, for gravity for example, that there are several models out there that are consistent with observations, but we do not know which ones can be consistently included in a fundamental theory.

And theory gives us ideas of what to look for. In this thread "seeing extra dimensions" are discussed, but it is misrepresented a bit. There are potentially several ways that we could start seeing evidence for extra dimensions, at least in principle. For example, "compactification" which means that we make the extra dimensions small, hidden for us, comes with the so-called "Kaluza-Klein tower" of particles, in which particles are essentially separated in mass inversely to the size of the size of the extra dimensions (small extra dimensions -> high mass). So this is one indirect way of how one could in principle see them (then they may be very massive, and virtually undiscoverable, but space-time warping brings down these masses... so we don't know).

Some of the particle physics experiments are looking for, in a sense, "anything that deviates" from the standard model. Note that for such experiments, any fundamental theory would have the same "problem" as string theory: it must show new physics at higher energies than already explored. LHC results are often seen as a "string theory is wrong"-result, which is not true, but what it rather shows is how boring the universe is at those energies, independently of the theory. Hopefully theory can give predictions in other places as well (in addition to the predictions of susy, extra dimensions, etc), like of what gravitational waves can say something fundamental about black holes.