Ok, let me say a word about spinors. They first were introduced in Physics to describe something quantum, discrete. Note, geometry has also been introduced in Physics, in Classical Physics, to be exact. The Classical Physics deals with inclusive pictures: any image contains many "pixels" (points) taken together. The number of points must be large enough to create a clear continuous image, but not too many in order not to burn the human eye. And the number of points must not be too small in order not to produce a dim and uncertain image. These numbers are governed with inequalities. But after inventing mathematics (geometry), those inequalities were forgotten. In Mathematics the Sun shines as one likes ;-). Continuity is taken as granted, but it is an unjustified extrapolation.
When we take too few points, we get into the realm of Quantum Mechanics. The truth is that we "obtain" information from elementary (indivisible) points. You may find many double slit experiment cartoons showing how the interference image is progressively built from collecting points.
Now vectors. Vectors are classical objects. They may have any length $L$. As you may now guess, the "continuous" character of vector length is illusory - it consists, in Physics, of many elementary pieces taken together. Trying to consider shorter and shorter vectors in Physics encounters the same effect: there is the smallest length (I am here speaking of angular momentum vector). Thus spinors were introduced, with differnet rules (QM) than vectors (CM).
It is not really correct to say that $1/2\otimes 1/2=0\oplus 1$ means obtaining a vector from spinors. Because such a product is of a certain length $L=\sqrt{1\cdot(1+1)}=\sqrt{2}$ in $\hbar$ units and it has three discrete projections on the $z$-axis. It is a spin-1 QM state, not a vector in a classical sense! A vector in a classical sense - a quantity of arbitrary length and projections - can be obtained as a product of many-many spinors (I omit how one gets a vector rather than a tensor of a higher rank from such a product, see QM textbooks for that), and, maybe, from a superposition of such spinor products, and again, as an inclusive quantity.
About rotations: In Physics rotation transformations describe recalculation rules of the results obtained in one reference frame into another RF. They do not describe rotations of the object itself. There is no rotation velocity involved in such transformations: the group $SO(3)$ contains three angles and nothing else. And the most important thing in this recalculation rules is that they only apply to the mean (average) values, not to the separate dots. For example, a photon spontaneously emitted with a resting excited atom to the left will be registered with the left photo-detector, and no recalculation rules may "transform" this signal to a zero signal of the right photo-detector. Only average (mean) results of a series of similar experiments are connected with "rotational transformations".
So I think your questions stem from "too mathematical" perception of objects introduced in Physics and forgetting the inequalities limiting validity of mathematical abstractions for physical notions.
One of implicit errors is assigning a spin to a "spatially localized particle". If you read Ohanian about spin, you will see that it is the boundary conditions (BC) that form the wave packet. In other words, a spin 1/2 electron is a quasi-particle of a certain huge system (replaced with some BC - for mathematical simplicity), which determines its properties.
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