The answer to all questions is No. In fact, even the right reaction to the first sentence - that the Planck scale is a "discrete measure" - is No.
The Planck length is a particular value of distance which is as important as $2\pi$ times the distance or any other multiple. The fact that we can speak about the Planck scale doesn't mean that the distance becomes discrete in any way. We may also talk about the radius of the Earth which doesn't mean that all distances have to be its multiples.
In quantum gravity, geometry with the usual rules doesn't work if the (proper) distances are thought of as being shorter than the Planck scale. But this invalidity of classical geometry doesn't mean that anything about the geometry has to become discrete (although it's a favorite meme promoted by popular books). There are lots of other effects that make the sharp, point-based geometry we know invalid - and indeed, we know that in the real world, the geometry collapses near the Planck scale because of other reasons than discreteness.
Quantum mechanics got its name because according to its rules, some quantities such as energy of bound states or the angular momentum can only take "quantized" or discrete values (eigenvalues). But despite the name, that doesn't mean that all observables in quantum mechanics have to possess a discrete spectrum. Do positions or distances possess a discrete spectrum?
The proposition that distances or durations become discrete near the Planck scale is a scientific hypothesis and it is one that may be - and, in fact, has been - experimentally falsified. For example, these discrete theories inevitably predict that the time needed for photons to get from very distant places of the Universe to the Earth will measurably depend on the photons' energy.
The Fermi satellite has showed that the delay is zero within dozens of milliseconds
http://motls.blogspot.com/2009/08/fermi-kills-all-lorentz-violating.html
which proves that the violations of the Lorentz symmetry (special relativity) of the magnitude that one would inevitably get from the violations of the continuity of spacetime have to be much smaller than what a generic discrete theory predicts.
In fact, the argument used by the Fermi satellite only employs the most straightforward way to impose upper bounds on the Lorentz violation. Using the so-called birefringence,
http://arxiv.org/abs/1102.2784
one may improve the bounds by 14 orders of magnitude! This safely kills any imaginable theory that violates the Lorentz symmetry - or even continuity of the spacetime - at the Planck scale. In some sense, the birefringence method applied to gamma ray bursts allows one to "see" the continuity of spacetime at distances that are 14 orders of magnitude shorter than the Planck length.
It doesn't mean that all physics at those "distances" works just like in large flat space. It doesn't. But it surely does mean that some physics - such as the existence of photons with arbitrarily short wavelengths - has to work just like it does at long distances. And it safely rules out all hypotheses that the spacetime may be built out of discrete, LEGO-like or any qualitatively similar building blocks.
Best Answer
In the usual definition of general relativity, spacetime is continuous. However, general relativity is a classical theory and does not take quantum effects into account. Such effects are expected to show up at very short distances, where your question is relevant.
All the experimental evidence points to continuous space, down to the shortest distances at which we have been able to measure. We don't know what happens at shorter distances. We also do not have any direct experimental evidence that gravity is a quantum theory, with the same caveat.
On the other hand, we are quite confident that a complete theory of nature must include quantum gravity and not just classical gravity. And, we have an educated guess of the distance scale at which quantum effects should become measurable: this is the Planck length, roughly $10^{-33}$ cm. This is much much shorter than the shortest distance at which we can carry out experiments, so at least we are not surprised that we did not see any such effects so far.
Before proceeding, one more caveat. There is an interesting and quite recent astrophysical experiment that showed that Lorentz symmetry holds even below the Planck length. If Lorentz symmetry is broken, it generally means that photons with different energies will travel at different velocities. At the experiment, they managed to detect a pair of photons that were created at almost the same time but had very different energies. They reached the detector almost simultaneously, which means their velocities were similar. Because the photons travelled an enormous distance before reaching us, they must have had almost the same velocity.
So we know that at least Lorentz symmetry holds at very short distances, and it seems difficult to reconcile this experimental fact with a discrete spacetime. So at least naively it seems that this is evidence against discreteness.
At long distances spacetime can certainly be thought of as continuous. At short distances, the short answer is: we don't know.
String theory is the only consistent theory of quantum gravity we know of, where we can actually compute things with some confidence. (You will probably hear some opinions that contradict this statement, mentioning loop quantum gravity, causal sets, etc., which are not related to string theory, but what I said is the common view in the community of high-energy theorists.) String theory is giving us some strong hints that perhaps spacetime at short distances is not continuous or discrete, but something else that we don't understand yet.
So the situation is that even theoretically, without talking about actual experiments that check the theory, we don't know what spacetime is like at short distances. Perhaps this is why you don't see this question mentioned a lot. My personal guess is that spacetime at short distances is neither continuous nor discrete, but has a different nature that may require new mathematical tools to describe.
Adding extra dimensions does not change any of the above.