I seem to be confused on a very particular note regarding CW complexes. Given the standard definition for such a complex (e.g. Hatcher's), would gluing $S^2$, for instance, along only the upper hemisphere of $S^1$, constitute a CW complex?
The root of this question, in general is as follows: does the characteristic map of a cell, attaching $\partial D^n_\alpha$ to the skeleton $X^{n-1}$, need to neccesarily contain entire $k$-cells ($k<n$) in its image?
To answer your question in general: No, the attaching map $\partial D^n_\alpha \to X^{n-1}$ can be any continuous function at all.
Your particular example is hard to understand. If your continuous map from $S^2$ to the upper hemisphere of $S^1$ is intended to be the attaching map of a 3-cell $D^3_\alpha$ then the answer is: yes, that can work.