I'll start with just a couple of ultra-simple points, and pass the question along to a knowledgeable theorist (Rob Leigh) for help with everything that requires any specialized knowledge.
A) I think you've got a little misunderstanding about what it means for something to be "N-dimensional". A point, with no extension in any direction, is zero
-dimensional. A line, as you say, is one-dimensional. So is a curvy line, or a line that loops back on itself. No finite bends or twists change the basic dimensionality, which just tells you haw many directions of extension are needed to describe a little piece. There are several more formal definitions of dimensionality, in agreement with each other for non-pathological objects, but this should suffice for now. You can see that a 2-D object could then be a flat sheet, a wavy sheet, the surface of a balloon, the surface of a donut, etc.
So 1-D or 2-D objects don't have 3-D volume, but they do have "extension".
B) Anything above 0-D can have some sort of vibrations, in the sense that some sort of physically defined distances between objects on it can change in time. Rob elaborates on this point: "The confusion here is between the dimension of the object and the dimension of the space that it is embedded in. Correspondingly there is a distinction between intrinsic geometry (e.g. a circle has no intrinsic curvature) and extrinsic geometry (e.g. a circle drawn in, say, 3d, has extrinsic curvature (1/radius)). In string theory, a simple interpretation is that loops are embedded in d-dimensional space (d=9, say), and they propagate in time. As time goes on, the loop sweeps out a world-sheet in space-time (much like a (0-dimensional) particle sweeps out a world-line in space-time)."
C) I'll hold off on discussing your point (C) until Rob can help with the description of the connection between the 1-D extensions of the string and the 3+1-D spacetime in which we picture things.
And Rob comes through:"Here's a picture of a simple string interaction that I stole from wikipedia.
The interpretation of this is of the string world-sheet embedded in 2+1 space-time (for visualization purposes). You should think of time as progressing in the downward direction. A string is obtained by slicing horizontally at constant time. So initially there is one string loop and it splits into two string loops. The correspondence with particle physics is that small strings correspond to light (in mass) particles. Thus if you take this diagram and shrink the radii of the loops down, it will start to look like a Feynman diagram for pair creation. This interpretation does require that the strings are embedded in and propagate in space-time (although technically this is just an interpretation of a theory that exists independently of this interpretation, but that is beyond the scope). "
(published on 03/10/2011)