Does space curvature automatically imply extra dimensions?

No, general relativity is based on something called "intrinsic curvature", which is related to how much parallel lines deviate towards or away from each other. It doesn't require embedding space-time in a higher dimensional structure to work. You're thinking of something called "extrinsic curvature". In fact, many examples of extrinsic curvature - including your example of a stick being bent - don't have intrinsic curvature at all. Let me try to be a bit clearer: imagine there is an ant who lives on your stick. As far as the ant is concerned, the world is one dimensional. Now, suppose we tell the ant that space is really 3D and his little 1D world is inside ("embedded in") that 3D space. There is absolutely no way the ant would be able to figure out if his stick was straight or bent the way you're describing. So, this isn't the sort of curvature that interests us in general relativity.

Basically, intrinsic curvature is just concerned with the geometric relationship between nearby points. It's entirely possible to think about this in terms of embedding space-time into some higher dimensional world, but you don't have to: it works just fine if you confine yourself to the observable four space-time dimensions. "Curved" in this sense is just a short hand way of saying "parallel lines don't do what they do in 'flat' (Euclidean or Minkowski) space / space-time".

Nope, spacetime curvature says nothing about the dimensionality. Your intuition here is probably wrong because human imagination needs 'some dimension to bend into' in order for something to be curved (i.e. an embedding in a higher-dimensional space). This is just our lack of imagination showing, though.