What is the name of a fully-twisted strip? I know that an odd number of twists generate a Möbius strip, and that an even number of twists generate a shape with two sides. Now, as far as I could find, for the strip with an even number of twists it's only said that the figure is topologically homeomorphic to an ordinary cylinder.
What is the name of this figure? And if there is no name, why is that so?
 A: As you say, a strip with an odd number of half-twists is a Möbius up to homeomorphism, and one with an even number of strips is an annulus up to homeomorphism. So from that point of view, we only have the two cases. I mention this because you seem happy with the same term for $1$ and $3$ half-twists, but want different terms for $0$ an $2$ half-twists.
When embedded in $\mathbb R^3$, different numbers of twists of course give non-equivalent surfaces in another sense (specifacally, they are not ambient isotopic, or even isotopic). But in this case, I've always only heard it as a strip/band/annulus with a given number of half-twists or full twists. So your shape would just be a "band with a full twist", or similar.
Or maybe someone knows of a specific term for it.
A: You are probably familiar with the language of knots/links. A knot is a circle embedded in $S^3$ (or $\mathbb{R}^3$), considered up to isotopy of the ambient space. Now, for each integer $n$, there is a distinct framing of the unknot, which you can picture as embedding of a ribbon $([0,1] \times S^1 \hookrightarrow S^3)$ rather than a string $(S^1 \hookrightarrow S^3)$.
As you correctly observed, each framed unknot is homeomorphic to either the cylinder or Möbius strip, depending on parity. Thus, any pair of these are homeomorphic to one another if they have the same parity. But they are not ambient isotopic (i.e. you can't deform one into another without cutting and gluing, which is of course discontinuous).
