These are really good questions, particularly as they were formed based on knowledge gleaned from the method most people these days get their initial exposure to science concepts - movies. I'll answer each of them - some are more complex than others.
- Einstein (or perhaps it was his colleague de Sitter) considered time as an extra dimension, which could only go in one direction (time's arrow), and used this to model concepts of special relativity. But it was considered an entire dimension, because you could (either through measurement of what happened or prediction of what will happen) determine events both past and in the future.
But the idea of partial dimensions (e.g. a dimension of value 0.5) is not silly. Fractals use partial dimensions to determine complexity (roughness) of surfaces and contours. So there may be a possibility of obtaining an object with pi dimensions. In holography, the third dimension is obtained by the phase of the light source, which is a limited dimension (i.e. it contains only information about the incident surfaces of objects, not about the interior or surfaces in the shadow of the object). Further, current physicists such as Susskind consider the entire universe to be utilising the holographic principle, so there is potential for partial dimensions to enter the lexicon of cosmology as well. But at the moment, I am unaware of partial dimensions in cosmology, but that does not mean it is a bad idea.
There are several theories of multiple universes and parallel worlds. But dimensions that you can't see (as in the 10 or 11 dimension theories) are a different beast - these dimensions are less than 1mm long, and are looped back into themselves. I believe that the parallel worlds idea that you are thinking of (like the sort in the television series Sliders) is a very different interpretation of quantum mechanics to ones that use unseen (parallel) dimensions (such as string and brane theories and quantum loop theories).
A singularity is a mathematical concept where things become infinite. The maths that predicts black holes puts a singularity at the center of the black hole, but a black hole is more than the singularity - it also has a Schwarzschild radius, from which even light cannot escape, and a whole bunch of other features. But the singularity is a problem to physicists - physics breaks down at singularities (there is no ability to predict therefore understand) so many believe that the theory is simply incomplete where singularities occur. When they find the right theory, everything else will be the same about the black hole, but the singularity won't exist in this new theory, and something will have taken its place (with perhaps a whole lot of other implications).